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{{About||the book by Ptolemy|Analemma (Ptolemy)|its more generic meaning as a graphical procedure|Orthographic projection}}
{{More citations needed|date=January 2009}}
[[File:Analemma fishburn.tif|thumb|upright=1.25|Afternoon analemma photo taken in 1998–99 in Murray Hill, New Jersey, USA, by Jack Fishburn. The Bell Laboratories building is in the foreground.]]
[[File:Globenmuseum Vienna 20091010 479.JPG|thumb|Analemma with date marks, printed on a globe, [[Globe Museum]], Vienna, Austria]]
In [[astronomy]], an '''analemma''' ({{IPAc-en|ˌ|æ|n|ə|ˈ|l|ɛ|m|ə}}; from [[Ancient Greek language|Greek]] {{lang|grc|ἀνάλημμα}} ''analēmma'' "support"){{efn| The word is rare in English, not to be found in most dictionaries. The Greek plural would be '''analemmata''', but in English '''analemmas''' is more frequently used.}} is a [[diagram]] showing the [[position of the Sun]] in the [[sky]] as seen from a fixed location on [[Earth]] at the same [[Solar time#Mean solar time|mean solar time]], as that position varies over the course of a [[year]]. The diagram will resemble a figure eight. [[Globe]]s of Earth often display an analemma as a 2- dimensional figure of [[equation of time]] vs. [[declination]] of the Sun.
The north–south component of the analemma results from the change in the Sun's [[declination]] due to the [[axial tilt|tilt]] of Earth's [[Rotation around a fixed axis|axis of rotation]]. The east–west component results from the [[equation of time|nonuniform rate of change]] of the Sun's [[right ascension]], governed by combined effects of Earth's axial tilt and [[orbital eccentricity]].
One can [[Photography|photograph]] an analemma by keeping a camera at a fixed location and orientation and taking multiple exposures throughout the year, always at the same [[time of day]] (disregarding [[daylight saving time]], if applicable).
Diagrams of analemmas frequently carry marks that show the position of the Sun at various closely spaced dates throughout the year. Analemmas with date marks can be used for various practical purposes.
Analemmas (as they are known today) have been used in conjunction with [[sundial]]s since the 18th century to convert between apparent and mean solar time. Before this, the term has a more generic meaning that refers to a graphical procedure of representing [[Three-dimensional space|three-dimensional]] objects in [[Two-dimensional space|two dimensions]], now known as [[orthographic projection]].<ref>{{cite journal|last=Sawyer|first=Frederick|title= Of Analemmas, Mean Time and the Analemmatic Sundial - Part 1|journal=Bulletin of the British Sundial Society|volume=6|issue=2|date =June 1994|pp=2-6|url=https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1994/Bull094-2.pdf#page=2}}</ref><ref>{{cite journal|last=Sawyer|first=Frederick|title= Of Analemmas, Mean Time and the Analemmatic Sundial - Part 2|journal=Bulletin of the British Sundial Society|volume=7|issue=1|date =February 1995|pp=39-44|url=https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1995/Bull095-1.pdf#page=39}}</ref>
Although the term ''analemma'' usually refers to Earth's solar analemma, it can be applied to other [[astronomical object|celestial bodies]] as well.
== Description ==
An analemma can be traced by plotting the position of the Sun as viewed from a fixed position on Earth at the same clock time every day for an entire year, or by plotting a graph of the Sun's [[declination]] against the [[equation of time]]. The resulting curve resembles a long, slender figure-eight with one lobe much larger than the other. This curve is commonly printed on [[Globe#Terrestrial and planetary|terrestrial globes]], usually in the eastern Pacific Ocean, the only large tropical region with very little land. It is possible, though challenging, to photograph the analemma, by leaving the camera in a fixed position for an entire year and snapping images on 24-hour intervals (or some multiple thereof); see section below.
The long axis of the figure—the line segment joining the northernmost point on the analemma to the southernmost—is [[Bisection|bisected]] by the [[celestial equator]], to which it is approximately [[perpendicular]], and has a "length" of twice the [[Axial tilt|obliquity of the ecliptic]], i.e., about 47°. The component along this axis of the Sun's apparent motion is a result of the familiar seasonal variation of the [[declination]] of the Sun through the year. The "width" of the figure is due to the equation of time, and its angular extent is the difference between the greatest positive and negative deviations of [[local solar time]] from [[local mean time]] when this time-difference is related to angle at the rate of 15° per hour, i.e., 360° in 24 h. This width of the analemma is approximately 7.7°, so the length of the figure is more than six times its width. The difference in size of the lobes of the figure-eight form arises mainly from the fact that the [[perihelion and aphelion]] occur far from [[equinox]]es. They also occur a mere couple of weeks after [[solstice]]s, which in turn causes slight tilt of the figure eight and its minor lateral asymmetry.
There are three parameters that affect the size and shape of the analemma—[[axial tilt|obliquity]], [[orbital eccentricity|eccentricity]], and the angle between the [[apse line]] and the line of [[solstice]]s. Viewed from an object with a perfectly circular [[orbit]] and no axial tilt, the Sun would always appear at the same point in the sky at the same time of day throughout the year and the analemma would be a dot. For an object with a circular orbit but significant axial tilt, the analemma would be a figure of eight with northern and southern lobes equal in size. For an object with an eccentric orbit but no axial tilt, the analemma would be a straight east–west line along the celestial equator.
The north–south component of the analemma shows the [[Position of the Sun#Declination of the Sun as seen from Earth|Sun's declination]], its latitude on the celestial sphere, or the latitude on the Earth at which the Sun is directly overhead. The east–west component shows the [[equation of time]], or the difference between [[solar time]] and [[local mean time]]. This can be interpreted as how "fast" or "slow" the Sun (or an [[analemmatic sundial]]) is compared to clock time. It also shows how far west or east the Sun is, compared with its mean position. The analemma can be considered as a graph in which the Sun's declination and the equation of time are plotted against each other. In many diagrams of the analemma, a third dimension, that of time, is also included, shown by marks that represent the position of the Sun at various, fairly closely spaced, dates throughout the year.
In diagrams, the analemma is drawn as it would be seen in the sky by an observer looking upward. If north is at the top, ''west'' is to the ''right''. This corresponds with the sign of the equation of time, which is positive in the westward direction. The further west the Sun is, compared with its mean position, the more "fast" a sundial is, compared with a clock. (See [[Equation of time#Sign of the equation of time]].) If the analemma is a graph with positive declination (north) plotted upward, positive equation of time (west) is plotted to the right. This is the conventional orientation for graphs. When the analemma is marked on a geographical globe, west in the analemma is to the right, while the geographical features on the globe are shown with west to the left. To avoid this confusion, it has been suggested that analemmas on globes should be printed with west to the left, but this is not done, at least, not frequently. In practice, the analemma is so nearly symmetrical that the shapes of the mirror images are not easily distinguished, but if date markings are present, they go in opposite directions. The Sun moves eastward on the analemma near the solstices. This can be used to tell which way the analemma is printed. See the image above, [http:/upwiki/wikipedia/commons/thumb/6/6d/Globenmuseum_Vienna_20091010_479.JPG/1280px-Globenmuseum_Vienna_20091010_479.JPG at high magnification].
An analemma that includes an image of a solar eclipse is called a '''tutulemma''', a term coined by photographers Cenk E. Tezel and [[Tunç Tezel]] based on the Turkish word for eclipse.<ref>{{Cite APOD |date=20 December 2009 |title=Tutulemma: Solar Eclipse Analemma |access-date=}}</ref>
==As seen from Earth==
[[File:Analemma on earth globe.png|thumb|Analemma on Earth as the position of the sun is directly overhead every 24 hours over one year.]]
[[File:Analemma Earth.png|thumb|left|upright=1.5|Analemma plotted as seen at noon GMT from the [[Royal Observatory, Greenwich]] ([[latitude]] 51.48° north, [[longitude]] 0.0015° west).]]
Owing to the tilt of Earth's axis (23.439°) and the Earth's orbital eccentricity, the relative location of the Sun above the horizon is not constant from day to day when observed at the same clock time each day. If the time of observation is not 12:00 noon local mean time, then depending on one's geographical latitude, this loop will be inclined at different angles.
The figure on the left is an example of an analemma as seen from the Earth's [[northern hemisphere]]. It is a plot of the position of the Sun at 12:00 noon at [[Royal Observatory, Greenwich]], England ([[latitude]] 51.48°N, [[longitude]] 0.0015°W) during the year 2006. The horizontal axis is the [[azimuth]] angle in degrees (180° is facing south). The vertical axis is the [[altitude (astronomy)|altitude]] in degrees above the horizon. The first day of each month is shown in black, and the [[solstice]]s and [[equinox]]es are shown in green. It can be seen that the equinoxes occur approximately at altitude {{nowrap|1=''φ'' = 90° − 51.5° = 38.5°}}, and the solstices occur approximately at altitudes {{nowrap|''φ'' ± ''ε''}} where ''ε ''is the [[axial tilt]] of the earth, 23.4°. The analemma is plotted with its width highly exaggerated, revealing a slight asymmetry (due to the two-week misalignment between the [[Apsis|apsides]] of the Earth's orbit and its [[solstice]]s).
The analemma is oriented with the smaller loop appearing north of the larger loop. At the [[North Pole]], the analemma would be completely upright (an 8 with the small loop at the top), and only the top half of it would be visible. Heading south, once south of the [[Arctic Circle]], the entire analemma would become visible. If you see it at noon, it continues to be upright, and rises higher from the horizon as you move south. When you get to the equator, it is directly overhead. As you go further south, it moves toward the northern horizon, and is then seen with the larger loop at the top. If, on the other hand, you looked at the analemma in the early morning or evening, it would start to tilt to one side as you moved southward from the North Pole. By the time you got to the [[equator]], the analemma would be completely horizontal. Then, as you continued to go south, it would continue rotating so that the small loop was beneath the large loop in the sky. Once you crossed the [[Antarctic Circle]], the analemma, now nearly completely inverted, would start to disappear, until only 50%, part of the larger loop, was visible from the [[South Pole]].<ref name="scienceblogs">[http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php Why Our Analemma Looks like a Figure 8] {{webarchive |url=https://web.archive.org/web/20120117120418/http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php |date=January 17, 2012 }}</ref>
See [[equation of time]] for a more detailed description of the east–west characteristics of the analemma.
==Photography==
The first successful analemma photograph ever made was created in 1978–79 by photographer [[Dennis di Cicco]] over [[Watertown, Massachusetts]]. Without moving his camera, he made 44 exposures on a single frame of film, all taken at the same time of day at least a week apart. A foreground image and three [[long-exposure photography|long-exposure images]] were also included in the same frame, bringing the total number of exposures to 48.<ref>{{cite news | title = More People Have Walked on the Moon Than Have Captured the Analemma | date = 20 September 2011 | url = https://petapixel.com/2011/09/20/more-people-have-walked-on-the-moon-than-have-captured-the-analemma/ | work = PetaPixel | access-date = 2017-07-06}} Includes image of original 1979 publication.</ref>
==Calculated analemmas==
[[File:Wreath_of_Analemmas.png|thumb|"Wreath of Analemmas". Analemmas calculated at 1-hour apart from each other for the geographic center of the contiguous United States. The gray part indicates it is nighttime.]]
While photographing analemmas may face technical and practical challenges, they could be calculated conveniently and presented in 3D plots for any given location on the surface of the Earth.<ref name="Zhangetal">{{cite journal | last=Zhang | first=Taiping | last2=Stackhouse | first2=Paul W. | last3=Macpherson | first3=Bradley | last4=Mikovitz | first4=J. Colleen | title=A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function | journal=Renewable Energy | publisher=Elsevier BV | volume=172 | year=2021 | issn=0960-1481 | doi=10.1016/j.renene.2021.03.047 | pages=1333–1340| doi-access=free }}</ref>
The idea is based on the unit vector with its origin fixed at a chosen point on the surface of the Earth and its direction pointing to the center of the Sun all the time. If we calculate the position of the Sun, namely, the [[solar zenith angle]] and [[solar azimuth angle]] at say, one-hour step, for an entire year, the head of the unit vector traces out 24 analemmas on the unit sphere centered on the chosen point, and this unit sphere is equivalent to the [[celestial sphere]]. The figure on the right is the "wreath of analemmas" calculated for the geographic center of the contiguous United States.
[[File:analemma_EoT_vs_Delta_2020.png|thumb|left| Analemma: Equation of time vs. declination of the Sun. Calculated for the year 2020 using the formulas from ''The [[Astronomical Almanac]] for the Year 2019''.]]
As often seen on a globe, the analemma is also often plotted as a two-dimensional figure of [[equation of time]] vs. declination of the Sun. The adjacent figure ("Analemma: Equation of time...") is calculated using the algorithm presented in the reference<ref name="Zhangetal" /> that uses the formulas given in ''The [[Astronomical Almanac]] for the Year 2019''.
==Estimating sunrise and sunset data==
If marked to show the position of the Sun on it at fairly regular intervals (such as the 1st, 11th, and 21st days of every [[calendar month]]) the analemma summarises the apparent motion of the Sun, relative to its mean position, throughout the [[tropical year|year]]. A date-marked diagram of the analemma, with equal scales in both [[north]]–[[south]] and [[east]]–[[west]] directions, can be used as a tool to estimate quantities such as the times of [[sunrise]] and [[sunset]], which depend on the Sun's position. Generally, making these estimates depends on visualizing the analemma as a rigid structure in the sky, which moves around the Earth at constant speed so it rises and sets once a day, with the Sun slowly moving around it once a year.
Some approximations are involved in the process, chiefly the use of a plane diagram to represent things on the celestial sphere, and the use of drawing and measurement instead of numerical calculation. Because of these, the estimates are not perfectly precise, but they are usually good enough for practical purposes. Also, they have instructional value, showing in a simple visual way how the times of sunrises and sunsets vary.
===Earliest and latest sunrise and sunset===
[[File:Analemma pattern in the sky.jpg|thumb|upright=1.4|Diagram of an analemma looking east in the [[Northern Hemisphere]]. The dates of the Sun's position are shown. This analemma is calculated for 9am, not photographed.]]
The analemma can be used to find the dates of the earliest and latest [[sunrise]]s and [[sunset]]s of the year. These do not occur on the dates of the [[solstice]]s.
With reference to the image of a simulated analemma in the eastern sky, the lowest point of the analemma has just risen above the horizon. If the Sun were at that point, sunrise would have just occurred. This would be the latest sunrise of the year, since all other points on the analemma would rise earlier. Therefore, the date of the latest sunrise is when the Sun is at this lowest point (29 December, when the analemma is tilted as seen from latitude 50° north, as is shown in the diagram); however, in some areas that use [[daylight saving time]], the date of the latest sunrise occurs on the day before daylight saving time ends. Similarly, when the Sun is at the highest point on the analemma, near its top-left end, (on 15 June) the earliest sunrise of the year will occur. Likewise, at sunset, the earliest sunset will occur when the Sun is at its lowest point on the analemma when it is close to the western horizon, and the latest sunset when it is at the highest point.
None of these points is exactly at one of the ends of the analemma, where the Sun is at a solstice. As seen from northern [[middle latitude]]s, as the diagram shows, the earliest sunset occurs some time before the December solstice – typically a week or two before it – and the latest sunrise happens a week or two after the solstice. Thus, the darkest evening occurs in early to mid-December, but the mornings keep getting darker until about the New Year.
[[File:Sunrise - Libreville, Gabon - 2008.svg|thumb|left|Graph of time of sunrise for [[Libreville]], [[Gabon]], which is very near the [[Equator]]. Note there are two maxima and two minima.]]
The exact dates are those on which the Sun is at the points where the horizon is [[tangent]]ial to the analemma, which in turn depend on how much the analemma, or the north–south meridian passing through it, is tilted from the vertical. This angle of tilt is essentially the co-latitude (90° minus the latitude) of the observer. Calculating these dates numerically is complex, but they can be estimated fairly accurately by placing a straight-edge, tilted at the appropriate angle, tangential to a diagram of the analemma, and reading the dates (interpolating as necessary) when the Sun is at the positions of contact.
In [[middle latitude]]s, the dates get further from the solstices as the absolute value of the latitude decreases. In near-equatorial latitudes, the situation is more complex. The analemma lies almost horizontal, so the horizon can be tangential to it at two points, one in each loop of the analemma. Thus there are two widely separated dates in the year when the Sun rises earlier than on adjoining dates, and so on.<ref>{{cite web |url=http://aa.usno.navy.mil/faq/docs/dark_days.php |title=The Dark Days of Winter}} at the [http://www.usno.navy.mil/USNO USNO website] {{webarchive |url=https://web.archive.org/web/20160131231447/http://www.usno.navy.mil/USNO |date=January 31, 2016 }}</ref>
===Times of sunrise and sunset===
A similar geometrical method, based on the analemma, can be used to find the times of [[sunrise]] and [[sunset]] at any place on Earth (except within or near the [[Arctic Circle]] or [[Antarctic Circle]]), on any date.
The [[origin of coordinates|origin]] of the analemma, where the solar [[declination]] and the [[equation of time]] are both zero, rises and sets at 6 a.m. and 6 p.m. [[local mean time]] on every day of the year, irrespective of the observer's [[latitude]]. (This estimation does not take account of [[atmospheric refraction]].) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude (as described above), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date (interpolating between the date markings as necessary), then at sunrise this line represents the horizon.
The origin [[diurnal motion|appears to move]] along the [[celestial equator]] at a speed of 15° per hour, the speed of the [[Earth's rotation]]. The distance along the celestial equator from the point where it intersects the horizon to the position of the origin of the analemma at sunrise is the distance the origin moves between 6 a.m. and the time of sunrise on the given date. Measuring the length of this equatorial segment therefore gives the difference between 6 a.m. and the time of sunrise.
The measurement should, of course, be done on the diagram, but it should be expressed in terms of the angle that would be subtended at an observer on the ground by the corresponding distance in the analemma in the sky. It can be useful to compare it with the length of the analemma, which subtends 47°. Thus, for example, if the length of the equatorial segment on the diagram is 0.4 times the length of the analemma on the diagram, then the segment in the celestial analemma would subtend 0.4 × 47° = 18.8° at the observer on the ground. The angle, in degrees, should be divided by 15 to get the time difference in hours between sunrise and 6 a.m. The sign of the difference is clear from the diagram. If the horizon line at sunrise passes above the origin of the analemma, the Sun rises before 6 a.m., and ''vice versa''.
The same technique can be used, ''[[mutatis mutandis]]'', to estimate the time of sunset. Note that the estimated times are in local mean time. Corrections must be applied to convert them to [[standard time]] or [[daylight saving time]]. These corrections will include a term that involves the observer's [[longitude]], so both the latitude and longitude affect the final result.
===Azimuths of sunrise and sunset===
The [[azimuth]]s (true [[compass]] bearings) of the points on the [[horizon]] where the Sun rises and sets can be easily estimated, using the same diagram as is used to find the times of [[sunrise]] and [[sunset]], as described above.
The point where the horizon intersects the [[celestial equator]] represents due east or west. The point where the Sun is at sunrise or sunset represents the direction of sunrise or sunset. Simply measuring the distance along the horizon between these points, in angular terms (comparing it with the length of the analemma, as described above), gives the angle between due east or west and the direction of sunrise or sunset. Whether the sunrise or sunset is north or south of due east or west is clear from the diagram. The larger loop of the analemma is at its southern end.
==Seen from other planets==
[[File:Mars analemma.GIF|thumb|An analemma as viewed from [[Mars]] ]]
On Earth, the analemma appears as a [[wikt:figure eight|figure-eight]], but on other [[Solar System]] bodies, it may be very different<ref>{{cite web |title=Other Analemmas |url=https://analemma.com/other-analemmas.html |website=analemma.com |access-date=24 March 2021}}</ref> due to the interplay between the three parameters determining the analemma: [[axial tilt]] of each body, [[orbital eccentricity|eccentricity]] of the body's [[elliptic orbit]], and position of either apses or equinoxes. Thus, if either of these variables (such as eccentricity) always dominates the other (as is the case on [[Mars]]), the analemma will resemble a [[Drop (liquid)|teardrop]]. If either of the variables (such as eccentricity) is significant, and the other is practically zero (as is the case on [[Jupiter]], with only a 3° tilt), the figure will be something much closer to an [[ellipse]]. If both are important enough, that sometimes eccentricity or axial tilt dominates, a figure-eight results.<ref name="scienceblogs" />{{Citation needed|reason=reference does not exist, the statements are not supported, and contradicted by calculations|date=January 2018}}
[[File:Mars Analemma Time Lapse Opportunity.webm|thumb|A [[time-lapse]] of an [[w:Analemma|analemma]] on [[w:Mars|Mars]]. Created using images of the [[w:MarsDial|MarsDial]] on the ''[[w:Opportunity (rover)|Opportunity]]'' rover.]]
In the following list, ''day'' and ''year'' refer to the [[synodic day]] and [[sidereal year]] of the particular body:
; [[Mercury (planet)|Mercury]]: Because [[orbital resonance]] makes the day exactly two years long, the method of plotting the Sun's position at the same time each day would yield only a single point. However, the [[equation of time]] can still be calculated for any time of the year, so an analemma can be graphed with this information. The resulting curve is a nearly straight east–west line.
; [[Venus]]: There are slightly less than two days per year, so it would take several years to accumulate a complete analemma by the usual method. The resulting curve is an ellipse.
; [[Mars]]: Teardrop.
; [[Jupiter]]: Ellipse.
; [[Saturn]]: Technically a figure-eight, but the northern loop is so small that it more closely resembles a teardrop.
; [[Uranus]]: Figure-eight. (Uranus is tilted past sideways to an angle of 98°. Its orbit is about as eccentric as Jupiter's and more eccentric than Earth's.)
; [[Neptune]]: Figure-eight.
{{clear}}
==Of geosynchronous satellites==
[[Image:Qzss-45-0.09.jpg|thumb|upright|Groundtrack of [[QZSS]] geosynchronous orbit. Seen from the ground, its analemma would have a similar shape.]]
[[Geosynchronous satellite]]s revolve around the Earth with a period of one [[sidereal day]]. Seen from a fixed point on the Earth's surface, they trace paths in the sky which repeat every day, and are therefore simple and meaningful analemmas. They are generally roughly elliptical, teardrop shaped, or figure-8 in shape. Their shapes and dimensions depend on the parameters of the orbits. A subset of geosynchronous satellites are [[geostationary satellites|geostationary ones]], which ideally have perfectly circular orbits, exactly in the Earth's equatorial plane. A geostationary satellite therefore ideally remains stationary relative to the Earth's surface, staying over a single point on the equator. No real satellite is exactly geostationary, so real ones trace small analemmas in the sky. Since the sizes of the orbits of geosynchronous satellites are similar to the size of the Earth, substantial [[parallax]] occurs, depending on the location of the observer on the Earth's surface, so observers in different places see different analemmas.
The paraboloidal dishes that are used for radio communication with geosynchronous satellites often have to move so as to follow the satellite's daily movement around its analemma. The mechanisms that drive them must therefore be programmed with the parameters of the analemma. Exceptions are dishes that are used with (approximately) geostationary satellites, since these satellites appear to move so little that a fixed dish can function adequately at all times.
[[File:Quasi-satellite diagram.png|thumb|left|upright|Orbital diagram of a quasi-satellite]]
==Of quasi-satellites==
A [[quasi-satellite]], such as the one shown in this diagram, moves in a [[Retrograde and prograde motion|prograde]] orbit around the Sun, with the same orbital period (which we will call a year) as the planet it accompanies, but with a different (usually greater) orbital eccentricity. It appears, when seen from the planet, to revolve around the planet once a year in the retrograde direction, but at varying speed and probably not in the ecliptic plane. Relative to its mean position, moving at constant speed in the ecliptic, the quasi-satellite traces an analemma in the planet's sky, going around it once a year.<ref name=analemma>{{cite journal | title = The analemma criterion: accidental quasi-satellites are indeed true quasi-satellites |first1=Carlos |last1=de la Fuente Marcos |last2=de la Fuente Marcos |first2=Raúl | journal = [[Monthly Notices of the Royal Astronomical Society]] | date = 2016 | volume = 462 | issue = 3 | pages = 3344–3349| arxiv = 1607.06686 | doi = 10.1093/mnras/stw1833 |bibcode = 2016MNRAS.462.3344D}}</ref>
{{clear}}
==See also==
<!-- Please keep entries in alphabetical order & add a short description per [[WP:SEEALSO]] -->
* ''[[Anathem]]''
* [[Armillary sphere]]
* ''[[De architectura]]''
* [[Epicycle]]
* [[Lemniscate]]
* ''[[On the Dioptra]]''
==References==
=== Footnotes ===
{{Notelist}}
===Citations===
{{Reflist}}
===Further reading===
{{Refbegin}}
* {{cite journal |bibcode=1972S&T....44...20O |title=The Shape of the Analemma |last1=Oliver |first1=Bernard M. |volume=44 |year=1972 |pages=20 |journal=Sky and Telescope}}
* {{cite journal |doi=10.1080/00038628.2004.9697037 |title=Analemma, the Ancient Sketch of Fictitious Sunpath Geometry—Sun, Time and History of Mathematics |year=2004 |last1=Kittler |first1=Richard |last2=Darula |first2=Stan |journal=Architectural Science Review |volume=47 |issue=2 |pages=141–4|s2cid=122005748 }}
* {{cite journal |doi=10.1111/j.1600-0498.2005.470304.x |title=Heron's Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities |year=2005 |last1=Sidoli |first1=Nathan |journal=Centaurus |volume=47 |issue=3 |pages=236–58|bibcode = 2005Cent...47..236S }}
* {{cite journal |doi=10.2151/jmsj.83.851 |title=On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere |year=2005 |last1=Semazzi |first1=Fredrick H.M. |last2=Scroggs |first2=Jeffrey S. |last3=Pouliot |first3=George A. |last4=McKee-Burrows |first4=Analemma Leia |last5=Norman |first5=Matthew |last6=Poojary |first6=Vikram |last7=Tsai |first7=Yu-Ming |journal=Journal of the Meteorological Society of Japan |volume=83 |issue=5 |pages=851–69|doi-access=free }}
* {{cite journal |doi=10.1002/asna.19272300202 |title=Das Analemma von Ptolemäus |trans-title=The analemma by Ptolemy |language=de |year=1927 |last1=Luckey |first1=P. |journal=Astronomische Nachrichten |volume=230 |issue=2 |pages=17–46 |bibcode=1927AN....230...17L}}
* {{cite journal |first=Yusif |last=Id |date=December 1969 |title=An Analemma Construction for Right and Oblique Ascensions |journal=The Mathematics Teacher |volume=62 |issue=8 |pages=669–72 |doi=10.5951/MT.62.8.0669 |jstor=27958259}}
* {{cite book |url=http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf |title=The Analemma for Latitudinally-Challenged People |first=Teo Shin |last=Yeow |year=2002 |type=BS Thesis |publisher=National University of Singapore |access-date=2006-02-05 |archive-url=https://web.archive.org/web/20110517111058/http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf |archive-date=2011-05-17 |url-status=dead }}
{{Refend}}
==External links==
{{wiktionary}}
{{Commons category|Analemma}}
* [http://www.perseus.gr/Astro-Solar-Analemma.htm Analemma Series from Sunrise to Sunset]
* [http://epod.usra.edu/blog/2005/01/colorado-analemma.html Earth Science Photo of the Day] (2005-01-22)
* [http://moonkmft.co.uk/EquationOfTime.html The Equation of Time and the Analemma] — by Kieron Taylor
* [http://www.nikolasschiller.com/blog/index.php/archives/2008/08/01/1449/ The Use of the Analemma] — from an inset from Bowles's New and Accurate Map of the World (1780)
* [http://www.astronomycorner.net/games/analemma.html Figure-Eight in the Sky] — contains link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities)
* [http://www.analemma.com/ Analemma.com] — dedicated to the analemma.
* [https://web.archive.org/web/20060323145857/http://www.wsanford.com/~wsanford/exo/sundials/analemma_calc.html Calculate and Chart the Analemma] — a web site offered by a [[Fairfax County Public Schools]] planetarium that describes the analemma and also offers a downloadable spreadsheet that allows the user to experiment with analemmas of varying shapes.
* [http://www.jgiesen.de/analemma/ Analemma Sundial Applet] — includes many reference charts.
* ''[http://demonstrations.wolfram.com/Analemmas/ Analemmas]'' — by [[Stephen Wolfram]] based on a program by Michael Trott, [[Wolfram Demonstrations Project]].
* ''[http://www.mail-archive.com/sundial@uni-koeln.de/msg11062.html Analemma in Verse]'' by Tad Dunne
* ''[http://www.spaceweather.com/glossary/tutulemma.htm The Making of a Tutulemma]'' by [[Tunç Tezel]]
* ''[http://analemma.pl/english-version Making of a Solargraphy Analemma]'' by [[Maciej Zapiór and Łukasz Fajfrowski]]
*[http://equation-of-time.info Equation-of-Time.info] - a multipage website with many illustrations and videos dedicated to the Equation of Time, its components, its history, how it can be displayed in tables, curves, analemmas, etc., its use to correct sundials, astronomy, clocks, how it can be produced mechanically and much more : by Kevin Karney
*[https://web.archive.org/web/20191018103653/https://ciechanow.ski/earth-and-sun/ Earth and Sun] — an interactive blog post explaining the phenomenon
*[[Astronomy Picture of the Day]]
** [http://antwrp.gsfc.nasa.gov/apod/ap020709.html 2002-07-09] — Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap030320.html 2003-03-20] — Sunrise Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap040621.html 2004-06-21] — Analemma over Ancient Nemea
** [http://antwrp.gsfc.nasa.gov/apod/ap050713.html 2005-07-13] — Analemma of the Moon
** [http://antwrp.gsfc.nasa.gov/apod/ap061223.html 2006-12-23] — Analemma over the Temple of Olympian Zeus
** [http://antwrp.gsfc.nasa.gov/apod/ap061230.html 2006-12-30] — Martian Analemma at Sagan Memorial Station (simulated)
** [http://antwrp.gsfc.nasa.gov/apod/ap070617.html 2007-06-17] — Analemma over Ukraine
** [http://antwrp.gsfc.nasa.gov/apod/ap071204.html 2007-12-04] — Analemma over New Jersey (film)
** [http://antwrp.gsfc.nasa.gov/apod/ap081221.html 2008-12-21] — Analemma over the Porch of Maidens
** [http://antwrp.gsfc.nasa.gov/apod/ap091220.html 2009-12-20] — Tutulemma: Solar Eclipse Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap101231.html 2010-12-31] — Analemma 2010
** [http://antwrp.gsfc.nasa.gov/apod/ap120920.html 2012-09-20] — Sunrise Analemma (with a little extra)
** [http://antwrp.gsfc.nasa.gov/apod/ap131014.html 2013-10-14] — High Noon Analemma Over Azerbaijan
** [http://antwrp.gsfc.nasa.gov/apod/ap140320.html 2014-03-20] — Solargraphy Analemma
{{Portal bar|Astronomy|Stars|Spaceflight|Outer space|Solar System}}
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<!-- [[Category:Astronomy]] too general -->
[[Category:Dynamics of the Solar System]]
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New page wikitext, after the edit (new_wikitext) | '{{Short description|Diagrammatic representation of sun's position over a period of time}}
{{About||the book by Ptolemy|Analemma (Ptolemy)|itsWIkIpedia sucksI display an analemma as a 2- dimensional figure of [[equation of time]] vs. [[declination]] of the Sun.
The north–south component of the analemma results from the change in the Sun's [[declination]] due to the [[axial tilt|tilt]] of Earth's [[Rotation around a fixed axis|axis of rotation]]. The east–west component results from the [[equation of time|nonuniform rate of change]] of the Sun's [[right ascension]], governed by combined effects of Earth's axial tilt and [[orbital eccentricity]].
One can [[Photography|photograph]] an analemma by keeping a camera at a fixed location and orientation and taking multiple exposures throughout the year, always at the same [[time of day]] (disregarding [[daylight saving time]], if applicable).
Diagrams of analemmas frequently carry marks that show the position of the Sun at various closely spaced dates throughout the year. Analemmas with date marks can be used for various practical purposes.
Analemmas (as they are known today) have been used in conjunction with [[sundial]]s since the 18th century to convert between apparent and mean solar time. Before this, the term has a more generic meaning that refers to a graphical procedure of representing [[Three-dimensional space|three-dimensional]] objects in [[Two-dimensional space|two dimensions]], now known as [[orthographic projection]].<ref>{{cite journal|last=Sawyer|first=Frederick|title= Of Analemmas, Mean Time and the Analemmatic Sundial - Part 1|journal=Bulletin of the British Sundial Society|volume=6|issue=2|date =June 1994|pp=2-6|url=https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1994/Bull094-2.pdf#page=2}}</ref><ref>{{cite journal|last=Sawyer|first=Frederick|title= Of Analemmas, Mean Time and the Analemmatic Sundial - Part 2|journal=Bulletin of the British Sundial Society|volume=7|issue=1|date =February 1995|pp=39-44|url=https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1995/Bull095-1.pdf#page=39}}</ref>
Although the term ''analemma'' usually refers to Earth's solar analemma, it can be applied to other [[astronomical object|celestial bodies]] as well.
== Description ==
An analemma can be traced by plotting the position of the Sun as viewed from a fixed position on Earth at the same clock time every day for an entire year, or by plotting a graph of the Sun's [[declination]] against the [[equation of time]]. The resulting curve resembles a long, slender figure-eight with one lobe much larger than the other. This curve is commonly printed on [[Globe#Terrestrial and planetary|terrestrial globes]], usually in the eastern Pacific Ocean, the only large tropical region with very little land. It is possible, though challenging, to photograph the analemma, by leaving the camera in a fixed position for an entire year and snapping images on 24-hour intervals (or some multiple thereof); see section below.
The long axis of the figure—the line segment joining the northernmost point on the analemma to the southernmost—is [[Bisection|bisected]] by the [[celestial equator]], to which it is approximately [[perpendicular]], and has a "length" of twice the [[Axial tilt|obliquity of the ecliptic]], i.e., about 47°. The component along this axis of the Sun's apparent motion is a result of the familiar seasonal variation of the [[declination]] of the Sun through the year. The "width" of the figure is due to the equation of time, and its angular extent is the difference between the greatest positive and negative deviations of [[local solar time]] from [[local mean time]] when this time-difference is related to angle at the rate of 15° per hour, i.e., 360° in 24 h. This width of the analemma is approximately 7.7°, so the length of the figure is more than six times its width. The difference in size of the lobes of the figure-eight form arises mainly from the fact that the [[perihelion and aphelion]] occur far from [[equinox]]es. They also occur a mere couple of weeks after [[solstice]]s, which in turn causes slight tilt of the figure eight and its minor lateral asymmetry.
There are three parameters that affect the size and shape of the analemma—[[axial tilt|obliquity]], [[orbital eccentricity|eccentricity]], and the angle between the [[apse line]] and the line of [[solstice]]s. Viewed from an object with a perfectly circular [[orbit]] and no axial tilt, the Sun would always appear at the same point in the sky at the same time of day throughout the year and the analemma would be a dot. For an object with a circular orbit but significant axial tilt, the analemma would be a figure of eight with northern and southern lobes equal in size. For an object with an eccentric orbit but no axial tilt, the analemma would be a straight east–west line along the celestial equator.
The north–south component of the analemma shows the [[Position of the Sun#Declination of the Sun as seen from Earth|Sun's declination]], its latitude on the celestial sphere, or the latitude on the Earth at which the Sun is directly overhead. The east–west component shows the [[equation of time]], or the difference between [[solar time]] and [[local mean time]]. This can be interpreted as how "fast" or "slow" the Sun (or an [[analemmatic sundial]]) is compared to clock time. It also shows how far west or east the Sun is, compared with its mean position. The analemma can be considered as a graph in which the Sun's declination and the equation of time are plotted against each other. In many diagrams of the analemma, a third dimension, that of time, is also included, shown by marks that represent the position of the Sun at various, fairly closely spaced, dates throughout the year.
In diagrams, the analemma is drawn as it would be seen in the sky by an observer looking upward. If north is at the top, ''west'' is to the ''right''. This corresponds with the sign of the equation of time, which is positive in the westward direction. The further west the Sun is, compared with its mean position, the more "fast" a sundial is, compared with a clock. (See [[Equation of time#Sign of the equation of time]].) If the analemma is a graph with positive declination (north) plotted upward, positive equation of time (west) is plotted to the right. This is the conventional orientation for graphs. When the analemma is marked on a geographical globe, west in the analemma is to the right, while the geographical features on the globe are shown with west to the left. To avoid this confusion, it has been suggested that analemmas on globes should be printed with west to the left, but this is not done, at least, not frequently. In practice, the analemma is so nearly symmetrical that the shapes of the mirror images are not easily distinguished, but if date markings are present, they go in opposite directions. The Sun moves eastward on the analemma near the solstices. This can be used to tell which way the analemma is printed. See the image above, [http:/upwiki/wikipedia/commons/thumb/6/6d/Globenmuseum_Vienna_20091010_479.JPG/1280px-Globenmuseum_Vienna_20091010_479.JPG at high magnification].
An analemma that includes an image of a solar eclipse is called a '''tutulemma''', a term coined by photographers Cenk E. Tezel and [[Tunç Tezel]] based on the Turkish word for eclipse.<ref>{{Cite APOD |date=20 December 2009 |title=Tutulemma: Solar Eclipse Analemma |access-date=}}</ref>
==As seen from Earth==
[[File:Analemma on earth globe.png|thumb|Analemma on Earth as the position of the sun is directly overhead every 24 hours over one year.]]
[[File:Analemma Earth.png|thumb|left|upright=1.5|Analemma plotted as seen at noon GMT from the [[Royal Observatory, Greenwich]] ([[latitude]] 51.48° north, [[longitude]] 0.0015° west).]]
Owing to the tilt of Earth's axis (23.439°) and the Earth's orbital eccentricity, the relative location of the Sun above the horizon is not constant from day to day when observed at the same clock time each day. If the time of observation is not 12:00 noon local mean time, then depending on one's geographical latitude, this loop will be inclined at different angles.
The figure on the left is an example of an analemma as seen from the Earth's [[northern hemisphere]]. It is a plot of the position of the Sun at 12:00 noon at [[Royal Observatory, Greenwich]], England ([[latitude]] 51.48°N, [[longitude]] 0.0015°W) during the year 2006. The horizontal axis is the [[azimuth]] angle in degrees (180° is facing south). The vertical axis is the [[altitude (astronomy)|altitude]] in degrees above the horizon. The first day of each month is shown in black, and the [[solstice]]s and [[equinox]]es are shown in green. It can be seen that the equinoxes occur approximately at altitude {{nowrap|1=''φ'' = 90° − 51.5° = 38.5°}}, and the solstices occur approximately at altitudes {{nowrap|''φ'' ± ''ε''}} where ''ε ''is the [[axial tilt]] of the earth, 23.4°. The analemma is plotted with its width highly exaggerated, revealing a slight asymmetry (due to the two-week misalignment between the [[Apsis|apsides]] of the Earth's orbit and its [[solstice]]s).
The analemma is oriented with the smaller loop appearing north of the larger loop. At the [[North Pole]], the analemma would be completely upright (an 8 with the small loop at the top), and only the top half of it would be visible. Heading south, once south of the [[Arctic Circle]], the entire analemma would become visible. If you see it at noon, it continues to be upright, and rises higher from the horizon as you move south. When you get to the equator, it is directly overhead. As you go further south, it moves toward the northern horizon, and is then seen with the larger loop at the top. If, on the other hand, you looked at the analemma in the early morning or evening, it would start to tilt to one side as you moved southward from the North Pole. By the time you got to the [[equator]], the analemma would be completely horizontal. Then, as you continued to go south, it would continue rotating so that the small loop was beneath the large loop in the sky. Once you crossed the [[Antarctic Circle]], the analemma, now nearly completely inverted, would start to disappear, until only 50%, part of the larger loop, was visible from the [[South Pole]].<ref name="scienceblogs">[http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php Why Our Analemma Looks like a Figure 8] {{webarchive |url=https://web.archive.org/web/20120117120418/http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php |date=January 17, 2012 }}</ref>
See [[equation of time]] for a more detailed description of the east–west characteristics of the analemma.
==Photography==
The first successful analemma photograph ever made was created in 1978–79 by photographer [[Dennis di Cicco]] over [[Watertown, Massachusetts]]. Without moving his camera, he made 44 exposures on a single frame of film, all taken at the same time of day at least a week apart. A foreground image and three [[long-exposure photography|long-exposure images]] were also included in the same frame, bringing the total number of exposures to 48.<ref>{{cite news | title = More People Have Walked on the Moon Than Have Captured the Analemma | date = 20 September 2011 | url = https://petapixel.com/2011/09/20/more-people-have-walked-on-the-moon-than-have-captured-the-analemma/ | work = PetaPixel | access-date = 2017-07-06}} Includes image of original 1979 publication.</ref>
==Calculated analemmas==
[[File:Wreath_of_Analemmas.png|thumb|"Wreath of Analemmas". Analemmas calculated at 1-hour apart from each other for the geographic center of the contiguous United States. The gray part indicates it is nighttime.]]
While photographing analemmas may face technical and practical challenges, they could be calculated conveniently and presented in 3D plots for any given location on the surface of the Earth.<ref name="Zhangetal">{{cite journal | last=Zhang | first=Taiping | last2=Stackhouse | first2=Paul W. | last3=Macpherson | first3=Bradley | last4=Mikovitz | first4=J. Colleen | title=A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function | journal=Renewable Energy | publisher=Elsevier BV | volume=172 | year=2021 | issn=0960-1481 | doi=10.1016/j.renene.2021.03.047 | pages=1333–1340| doi-access=free }}</ref>
The idea is based on the unit vector with its origin fixed at a chosen point on the surface of the Earth and its direction pointing to the center of the Sun all the time. If we calculate the position of the Sun, namely, the [[solar zenith angle]] and [[solar azimuth angle]] at say, one-hour step, for an entire year, the head of the unit vector traces out 24 analemmas on the unit sphere centered on the chosen point, and this unit sphere is equivalent to the [[celestial sphere]]. The figure on the right is the "wreath of analemmas" calculated for the geographic center of the contiguous United States.
[[File:analemma_EoT_vs_Delta_2020.png|thumb|left| Analemma: Equation of time vs. declination of the Sun. Calculated for the year 2020 using the formulas from ''The [[Astronomical Almanac]] for the Year 2019''.]]
As often seen on a globe, the analemma is also often plotted as a two-dimensional figure of [[equation of time]] vs. declination of the Sun. The adjacent figure ("Analemma: Equation of time...") is calculated using the algorithm presented in the reference<ref name="Zhangetal" /> that uses the formulas given in ''The [[Astronomical Almanac]] for the Year 2019''.
==Estimating sunrise and sunset data==
If marked to show the position of the Sun on it at fairly regular intervals (such as the 1st, 11th, and 21st days of every [[calendar month]]) the analemma summarises the apparent motion of the Sun, relative to its mean position, throughout the [[tropical year|year]]. A date-marked diagram of the analemma, with equal scales in both [[north]]–[[south]] and [[east]]–[[west]] directions, can be used as a tool to estimate quantities such as the times of [[sunrise]] and [[sunset]], which depend on the Sun's position. Generally, making these estimates depends on visualizing the analemma as a rigid structure in the sky, which moves around the Earth at constant speed so it rises and sets once a day, with the Sun slowly moving around it once a year.
Some approximations are involved in the process, chiefly the use of a plane diagram to represent things on the celestial sphere, and the use of drawing and measurement instead of numerical calculation. Because of these, the estimates are not perfectly precise, but they are usually good enough for practical purposes. Also, they have instructional value, showing in a simple visual way how the times of sunrises and sunsets vary.
===Earliest and latest sunrise and sunset===
[[File:Analemma pattern in the sky.jpg|thumb|upright=1.4|Diagram of an analemma looking east in the [[Northern Hemisphere]]. The dates of the Sun's position are shown. This analemma is calculated for 9am, not photographed.]]
The analemma can be used to find the dates of the earliest and latest [[sunrise]]s and [[sunset]]s of the year. These do not occur on the dates of the [[solstice]]s.
With reference to the image of a simulated analemma in the eastern sky, the lowest point of the analemma has just risen above the horizon. If the Sun were at that point, sunrise would have just occurred. This would be the latest sunrise of the year, since all other points on the analemma would rise earlier. Therefore, the date of the latest sunrise is when the Sun is at this lowest point (29 December, when the analemma is tilted as seen from latitude 50° north, as is shown in the diagram); however, in some areas that use [[daylight saving time]], the date of the latest sunrise occurs on the day before daylight saving time ends. Similarly, when the Sun is at the highest point on the analemma, near its top-left end, (on 15 June) the earliest sunrise of the year will occur. Likewise, at sunset, the earliest sunset will occur when the Sun is at its lowest point on the analemma when it is close to the western horizon, and the latest sunset when it is at the highest point.
None of these points is exactly at one of the ends of the analemma, where the Sun is at a solstice. As seen from northern [[middle latitude]]s, as the diagram shows, the earliest sunset occurs some time before the December solstice – typically a week or two before it – and the latest sunrise happens a week or two after the solstice. Thus, the darkest evening occurs in early to mid-December, but the mornings keep getting darker until about the New Year.
[[File:Sunrise - Libreville, Gabon - 2008.svg|thumb|left|Graph of time of sunrise for [[Libreville]], [[Gabon]], which is very near the [[Equator]]. Note there are two maxima and two minima.]]
The exact dates are those on which the Sun is at the points where the horizon is [[tangent]]ial to the analemma, which in turn depend on how much the analemma, or the north–south meridian passing through it, is tilted from the vertical. This angle of tilt is essentially the co-latitude (90° minus the latitude) of the observer. Calculating these dates numerically is complex, but they can be estimated fairly accurately by placing a straight-edge, tilted at the appropriate angle, tangential to a diagram of the analemma, and reading the dates (interpolating as necessary) when the Sun is at the positions of contact.
In [[middle latitude]]s, the dates get further from the solstices as the absolute value of the latitude decreases. In near-equatorial latitudes, the situation is more complex. The analemma lies almost horizontal, so the horizon can be tangential to it at two points, one in each loop of the analemma. Thus there are two widely separated dates in the year when the Sun rises earlier than on adjoining dates, and so on.<ref>{{cite web |url=http://aa.usno.navy.mil/faq/docs/dark_days.php |title=The Dark Days of Winter}} at the [http://www.usno.navy.mil/USNO USNO website] {{webarchive |url=https://web.archive.org/web/20160131231447/http://www.usno.navy.mil/USNO |date=January 31, 2016 }}</ref>
===Times of sunrise and sunset===
A similar geometrical method, based on the analemma, can be used to find the times of [[sunrise]] and [[sunset]] at any place on Earth (except within or near the [[Arctic Circle]] or [[Antarctic Circle]]), on any date.
The [[origin of coordinates|origin]] of the analemma, where the solar [[declination]] and the [[equation of time]] are both zero, rises and sets at 6 a.m. and 6 p.m. [[local mean time]] on every day of the year, irrespective of the observer's [[latitude]]. (This estimation does not take account of [[atmospheric refraction]].) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude (as described above), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date (interpolating between the date markings as necessary), then at sunrise this line represents the horizon.
The origin [[diurnal motion|appears to move]] along the [[celestial equator]] at a speed of 15° per hour, the speed of the [[Earth's rotation]]. The distance along the celestial equator from the point where it intersects the horizon to the position of the origin of the analemma at sunrise is the distance the origin moves between 6 a.m. and the time of sunrise on the given date. Measuring the length of this equatorial segment therefore gives the difference between 6 a.m. and the time of sunrise.
The measurement should, of course, be done on the diagram, but it should be expressed in terms of the angle that would be subtended at an observer on the ground by the corresponding distance in the analemma in the sky. It can be useful to compare it with the length of the analemma, which subtends 47°. Thus, for example, if the length of the equatorial segment on the diagram is 0.4 times the length of the analemma on the diagram, then the segment in the celestial analemma would subtend 0.4 × 47° = 18.8° at the observer on the ground. The angle, in degrees, should be divided by 15 to get the time difference in hours between sunrise and 6 a.m. The sign of the difference is clear from the diagram. If the horizon line at sunrise passes above the origin of the analemma, the Sun rises before 6 a.m., and ''vice versa''.
The same technique can be used, ''[[mutatis mutandis]]'', to estimate the time of sunset. Note that the estimated times are in local mean time. Corrections must be applied to convert them to [[standard time]] or [[daylight saving time]]. These corrections will include a term that involves the observer's [[longitude]], so both the latitude and longitude affect the final result.
===Azimuths of sunrise and sunset===
The [[azimuth]]s (true [[compass]] bearings) of the points on the [[horizon]] where the Sun rises and sets can be easily estimated, using the same diagram as is used to find the times of [[sunrise]] and [[sunset]], as described above.
The point where the horizon intersects the [[celestial equator]] represents due east or west. The point where the Sun is at sunrise or sunset represents the direction of sunrise or sunset. Simply measuring the distance along the horizon between these points, in angular terms (comparing it with the length of the analemma, as described above), gives the angle between due east or west and the direction of sunrise or sunset. Whether the sunrise or sunset is north or south of due east or west is clear from the diagram. The larger loop of the analemma is at its southern end.
==Seen from other planets==
[[File:Mars analemma.GIF|thumb|An analemma as viewed from [[Mars]] ]]
On Earth, the analemma appears as a [[wikt:figure eight|figure-eight]], but on other [[Solar System]] bodies, it may be very different<ref>{{cite web |title=Other Analemmas |url=https://analemma.com/other-analemmas.html |website=analemma.com |access-date=24 March 2021}}</ref> due to the interplay between the three parameters determining the analemma: [[axial tilt]] of each body, [[orbital eccentricity|eccentricity]] of the body's [[elliptic orbit]], and position of either apses or equinoxes. Thus, if either of these variables (such as eccentricity) always dominates the other (as is the case on [[Mars]]), the analemma will resemble a [[Drop (liquid)|teardrop]]. If either of the variables (such as eccentricity) is significant, and the other is practically zero (as is the case on [[Jupiter]], with only a 3° tilt), the figure will be something much closer to an [[ellipse]]. If both are important enough, that sometimes eccentricity or axial tilt dominates, a figure-eight results.<ref name="scienceblogs" />{{Citation needed|reason=reference does not exist, the statements are not supported, and contradicted by calculations|date=January 2018}}
[[File:Mars Analemma Time Lapse Opportunity.webm|thumb|A [[time-lapse]] of an [[w:Analemma|analemma]] on [[w:Mars|Mars]]. Created using images of the [[w:MarsDial|MarsDial]] on the ''[[w:Opportunity (rover)|Opportunity]]'' rover.]]
In the following list, ''day'' and ''year'' refer to the [[synodic day]] and [[sidereal year]] of the particular body:
; [[Mercury (planet)|Mercury]]: Because [[orbital resonance]] makes the day exactly two years long, the method of plotting the Sun's position at the same time each day would yield only a single point. However, the [[equation of time]] can still be calculated for any time of the year, so an analemma can be graphed with this information. The resulting curve is a nearly straight east–west line.
; [[Venus]]: There are slightly less than two days per year, so it would take several years to accumulate a complete analemma by the usual method. The resulting curve is an ellipse.
; [[Mars]]: Teardrop.
; [[Jupiter]]: Ellipse.
; [[Saturn]]: Technically a figure-eight, but the northern loop is so small that it more closely resembles a teardrop.
; [[Uranus]]: Figure-eight. (Uranus is tilted past sideways to an angle of 98°. Its orbit is about as eccentric as Jupiter's and more eccentric than Earth's.)
; [[Neptune]]: Figure-eight.
{{clear}}
==Of geosynchronous satellites==
[[Image:Qzss-45-0.09.jpg|thumb|upright|Groundtrack of [[QZSS]] geosynchronous orbit. Seen from the ground, its analemma would have a similar shape.]]
[[Geosynchronous satellite]]s revolve around the Earth with a period of one [[sidereal day]]. Seen from a fixed point on the Earth's surface, they trace paths in the sky which repeat every day, and are therefore simple and meaningful analemmas. They are generally roughly elliptical, teardrop shaped, or figure-8 in shape. Their shapes and dimensions depend on the parameters of the orbits. A subset of geosynchronous satellites are [[geostationary satellites|geostationary ones]], which ideally have perfectly circular orbits, exactly in the Earth's equatorial plane. A geostationary satellite therefore ideally remains stationary relative to the Earth's surface, staying over a single point on the equator. No real satellite is exactly geostationary, so real ones trace small analemmas in the sky. Since the sizes of the orbits of geosynchronous satellites are similar to the size of the Earth, substantial [[parallax]] occurs, depending on the location of the observer on the Earth's surface, so observers in different places see different analemmas.
The paraboloidal dishes that are used for radio communication with geosynchronous satellites often have to move so as to follow the satellite's daily movement around its analemma. The mechanisms that drive them must therefore be programmed with the parameters of the analemma. Exceptions are dishes that are used with (approximately) geostationary satellites, since these satellites appear to move so little that a fixed dish can function adequately at all times.
[[File:Quasi-satellite diagram.png|thumb|left|upright|Orbital diagram of a quasi-satellite]]
==Of quasi-satellites==
A [[quasi-satellite]], such as the one shown in this diagram, moves in a [[Retrograde and prograde motion|prograde]] orbit around the Sun, with the same orbital period (which we will call a year) as the planet it accompanies, but with a different (usually greater) orbital eccentricity. It appears, when seen from the planet, to revolve around the planet once a year in the retrograde direction, but at varying speed and probably not in the ecliptic plane. Relative to its mean position, moving at constant speed in the ecliptic, the quasi-satellite traces an analemma in the planet's sky, going around it once a year.<ref name=analemma>{{cite journal | title = The analemma criterion: accidental quasi-satellites are indeed true quasi-satellites |first1=Carlos |last1=de la Fuente Marcos |last2=de la Fuente Marcos |first2=Raúl | journal = [[Monthly Notices of the Royal Astronomical Society]] | date = 2016 | volume = 462 | issue = 3 | pages = 3344–3349| arxiv = 1607.06686 | doi = 10.1093/mnras/stw1833 |bibcode = 2016MNRAS.462.3344D}}</ref>
{{clear}}
==See also==
<!-- Please keep entries in alphabetical order & add a short description per [[WP:SEEALSO]] -->
* ''[[Anathem]]''
* [[Armillary sphere]]
* ''[[De architectura]]''
* [[Epicycle]]
* [[Lemniscate]]
* ''[[On the Dioptra]]''
==References==
=== Footnotes ===
{{Notelist}}
===Citations===
{{Reflist}}
===Further reading===
{{Refbegin}}
* {{cite journal |bibcode=1972S&T....44...20O |title=The Shape of the Analemma |last1=Oliver |first1=Bernard M. |volume=44 |year=1972 |pages=20 |journal=Sky and Telescope}}
* {{cite journal |doi=10.1080/00038628.2004.9697037 |title=Analemma, the Ancient Sketch of Fictitious Sunpath Geometry—Sun, Time and History of Mathematics |year=2004 |last1=Kittler |first1=Richard |last2=Darula |first2=Stan |journal=Architectural Science Review |volume=47 |issue=2 |pages=141–4|s2cid=122005748 }}
* {{cite journal |doi=10.1111/j.1600-0498.2005.470304.x |title=Heron's Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities |year=2005 |last1=Sidoli |first1=Nathan |journal=Centaurus |volume=47 |issue=3 |pages=236–58|bibcode = 2005Cent...47..236S }}
* {{cite journal |doi=10.2151/jmsj.83.851 |title=On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere |year=2005 |last1=Semazzi |first1=Fredrick H.M. |last2=Scroggs |first2=Jeffrey S. |last3=Pouliot |first3=George A. |last4=McKee-Burrows |first4=Analemma Leia |last5=Norman |first5=Matthew |last6=Poojary |first6=Vikram |last7=Tsai |first7=Yu-Ming |journal=Journal of the Meteorological Society of Japan |volume=83 |issue=5 |pages=851–69|doi-access=free }}
* {{cite journal |doi=10.1002/asna.19272300202 |title=Das Analemma von Ptolemäus |trans-title=The analemma by Ptolemy |language=de |year=1927 |last1=Luckey |first1=P. |journal=Astronomische Nachrichten |volume=230 |issue=2 |pages=17–46 |bibcode=1927AN....230...17L}}
* {{cite journal |first=Yusif |last=Id |date=December 1969 |title=An Analemma Construction for Right and Oblique Ascensions |journal=The Mathematics Teacher |volume=62 |issue=8 |pages=669–72 |doi=10.5951/MT.62.8.0669 |jstor=27958259}}
* {{cite book |url=http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf |title=The Analemma for Latitudinally-Challenged People |first=Teo Shin |last=Yeow |year=2002 |type=BS Thesis |publisher=National University of Singapore |access-date=2006-02-05 |archive-url=https://web.archive.org/web/20110517111058/http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf |archive-date=2011-05-17 |url-status=dead }}
{{Refend}}
==External links==
{{wiktionary}}
{{Commons category|Analemma}}
* [http://www.perseus.gr/Astro-Solar-Analemma.htm Analemma Series from Sunrise to Sunset]
* [http://epod.usra.edu/blog/2005/01/colorado-analemma.html Earth Science Photo of the Day] (2005-01-22)
* [http://moonkmft.co.uk/EquationOfTime.html The Equation of Time and the Analemma] — by Kieron Taylor
* [http://www.nikolasschiller.com/blog/index.php/archives/2008/08/01/1449/ The Use of the Analemma] — from an inset from Bowles's New and Accurate Map of the World (1780)
* [http://www.astronomycorner.net/games/analemma.html Figure-Eight in the Sky] — contains link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities)
* [http://www.analemma.com/ Analemma.com] — dedicated to the analemma.
* [https://web.archive.org/web/20060323145857/http://www.wsanford.com/~wsanford/exo/sundials/analemma_calc.html Calculate and Chart the Analemma] — a web site offered by a [[Fairfax County Public Schools]] planetarium that describes the analemma and also offers a downloadable spreadsheet that allows the user to experiment with analemmas of varying shapes.
* [http://www.jgiesen.de/analemma/ Analemma Sundial Applet] — includes many reference charts.
* ''[http://demonstrations.wolfram.com/Analemmas/ Analemmas]'' — by [[Stephen Wolfram]] based on a program by Michael Trott, [[Wolfram Demonstrations Project]].
* ''[http://www.mail-archive.com/sundial@uni-koeln.de/msg11062.html Analemma in Verse]'' by Tad Dunne
* ''[http://www.spaceweather.com/glossary/tutulemma.htm The Making of a Tutulemma]'' by [[Tunç Tezel]]
* ''[http://analemma.pl/english-version Making of a Solargraphy Analemma]'' by [[Maciej Zapiór and Łukasz Fajfrowski]]
*[http://equation-of-time.info Equation-of-Time.info] - a multipage website with many illustrations and videos dedicated to the Equation of Time, its components, its history, how it can be displayed in tables, curves, analemmas, etc., its use to correct sundials, astronomy, clocks, how it can be produced mechanically and much more : by Kevin Karney
*[https://web.archive.org/web/20191018103653/https://ciechanow.ski/earth-and-sun/ Earth and Sun] — an interactive blog post explaining the phenomenon
*[[Astronomy Picture of the Day]]
** [http://antwrp.gsfc.nasa.gov/apod/ap020709.html 2002-07-09] — Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap030320.html 2003-03-20] — Sunrise Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap040621.html 2004-06-21] — Analemma over Ancient Nemea
** [http://antwrp.gsfc.nasa.gov/apod/ap050713.html 2005-07-13] — Analemma of the Moon
** [http://antwrp.gsfc.nasa.gov/apod/ap061223.html 2006-12-23] — Analemma over the Temple of Olympian Zeus
** [http://antwrp.gsfc.nasa.gov/apod/ap061230.html 2006-12-30] — Martian Analemma at Sagan Memorial Station (simulated)
** [http://antwrp.gsfc.nasa.gov/apod/ap070617.html 2007-06-17] — Analemma over Ukraine
** [http://antwrp.gsfc.nasa.gov/apod/ap071204.html 2007-12-04] — Analemma over New Jersey (film)
** [http://antwrp.gsfc.nasa.gov/apod/ap081221.html 2008-12-21] — Analemma over the Porch of Maidens
** [http://antwrp.gsfc.nasa.gov/apod/ap091220.html 2009-12-20] — Tutulemma: Solar Eclipse Analemma
** [http://antwrp.gsfc.nasa.gov/apod/ap101231.html 2010-12-31] — Analemma 2010
** [http://antwrp.gsfc.nasa.gov/apod/ap120920.html 2012-09-20] — Sunrise Analemma (with a little extra)
** [http://antwrp.gsfc.nasa.gov/apod/ap131014.html 2013-10-14] — High Noon Analemma Over Azerbaijan
** [http://antwrp.gsfc.nasa.gov/apod/ap140320.html 2014-03-20] — Solargraphy Analemma
{{Portal bar|Astronomy|Stars|Spaceflight|Outer space|Solar System}}
{{Authority control}}
<!-- [[Category:Astronomy]] too general -->
[[Category:Dynamics of the Solar System]]
[[Category:Solar phenomena]]
[[Category:Sundials]]
[[Category:Articles containing video clips]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -1,9 +1,4 @@
{{Short description|Diagrammatic representation of sun's position over a period of time}}
-{{About||the book by Ptolemy|Analemma (Ptolemy)|its more generic meaning as a graphical procedure|Orthographic projection}}
-{{More citations needed|date=January 2009}}
-[[File:Analemma fishburn.tif|thumb|upright=1.25|Afternoon analemma photo taken in 1998–99 in Murray Hill, New Jersey, USA, by Jack Fishburn. The Bell Laboratories building is in the foreground.]]
-[[File:Globenmuseum Vienna 20091010 479.JPG|thumb|Analemma with date marks, printed on a globe, [[Globe Museum]], Vienna, Austria]]
-
-In [[astronomy]], an '''analemma''' ({{IPAc-en|ˌ|æ|n|ə|ˈ|l|ɛ|m|ə}}; from [[Ancient Greek language|Greek]] {{lang|grc|ἀνάλημμα}} ''analēmma'' "support"){{efn| The word is rare in English, not to be found in most dictionaries. The Greek plural would be '''analemmata''', but in English '''analemmas''' is more frequently used.}} is a [[diagram]] showing the [[position of the Sun]] in the [[sky]] as seen from a fixed location on [[Earth]] at the same [[Solar time#Mean solar time|mean solar time]], as that position varies over the course of a [[year]]. The diagram will resemble a figure eight. [[Globe]]s of Earth often display an analemma as a 2- dimensional figure of [[equation of time]] vs. [[declination]] of the Sun.
+{{About||the book by Ptolemy|Analemma (Ptolemy)|itsWIkIpedia sucksI display an analemma as a 2- dimensional figure of [[equation of time]] vs. [[declination]] of the Sun.
The north–south component of the analemma results from the change in the Sun's [[declination]] due to the [[axial tilt|tilt]] of Earth's [[Rotation around a fixed axis|axis of rotation]]. The east–west component results from the [[equation of time|nonuniform rate of change]] of the Sun's [[right ascension]], governed by combined effects of Earth's axial tilt and [[orbital eccentricity]].
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44 => 'http://antwrp.gsfc.nasa.gov/apod/ap131014.html',
45 => 'http://antwrp.gsfc.nasa.gov/apod/ap140320.html',
46 => 'http://demonstrations.wolfram.com/Analemmas/',
47 => 'http://epod.usra.edu/blog/2005/01/colorado-analemma.html',
48 => 'http://equation-of-time.info',
49 => 'http://moonkmft.co.uk/EquationOfTime.html',
50 => 'http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php',
51 => 'http:/upwiki/wikipedia/commons/thumb/6/6d/Globenmuseum_Vienna_20091010_479.JPG/1280px-Globenmuseum_Vienna_20091010_479.JPG',
52 => 'http://www.analemma.com/',
53 => 'http://www.astronomycorner.net/games/analemma.html',
54 => 'http://www.jgiesen.de/analemma/',
55 => 'http://www.mail-archive.com/sundial@uni-koeln.de/msg11062.html',
56 => 'http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf',
57 => 'http://www.nikolasschiller.com/blog/index.php/archives/2008/08/01/1449/',
58 => 'http://www.perseus.gr/Astro-Solar-Analemma.htm',
59 => 'http://www.spaceweather.com/glossary/tutulemma.htm',
60 => 'http://www.usno.navy.mil/USNO',
61 => 'https://academic.microsoft.com/v2/detail/112969597',
62 => 'https://analemma.com/other-analemmas.html',
63 => 'https://api.semanticscholar.org/CorpusID:122005748',
64 => 'https://apod.nasa.gov/apod/ap091220.html',
65 => 'https://d-nb.info/gnd/4766963-9',
66 => 'https://petapixel.com/2011/09/20/more-people-have-walked-on-the-moon-than-have-captured-the-analemma/',
67 => 'https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1994/Bull094-2.pdf#page=2',
68 => 'https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1995/Bull095-1.pdf#page=39',
69 => 'https://ui.adsabs.harvard.edu/abs/1927AN....230...17L',
70 => 'https://ui.adsabs.harvard.edu/abs/1972S&T....44...20O',
71 => 'https://ui.adsabs.harvard.edu/abs/2005Cent...47..236S',
72 => 'https://ui.adsabs.harvard.edu/abs/2016MNRAS.462.3344D',
73 => 'https://web.archive.org/web/20060323145857/http://www.wsanford.com/~wsanford/exo/sundials/analemma_calc.html',
74 => 'https://web.archive.org/web/20110517111058/http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf',
75 => 'https://web.archive.org/web/20120117120418/http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php',
76 => 'https://web.archive.org/web/20160131231447/http://www.usno.navy.mil/USNO',
77 => 'https://web.archive.org/web/20191018103653/https://ciechanow.ski/earth-and-sun/',
78 => 'https://www.jstor.org/action/doBasicSearch?Query=%22Analemma%22&acc=on&wc=on',
79 => 'https://www.wikidata.org/wiki/Q484737#identifiers'
] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Diagrammatic representation of sun's position over a period of time</div>
<p>{{About||the book by Ptolemy|Analemma (Ptolemy)|itsWIkIpedia sucksI display an analemma as a 2- dimensional figure of <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a> vs. <a href="/enwiki/wiki/Declination" title="Declination">declination</a> of the Sun.
</p><p>The north–south component of the analemma results from the change in the Sun's <a href="/enwiki/wiki/Declination" title="Declination">declination</a> due to the <a href="/enwiki/wiki/Axial_tilt" title="Axial tilt">tilt</a> of Earth's <a href="/enwiki/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">axis of rotation</a>. The east–west component results from the <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">nonuniform rate of change</a> of the Sun's <a href="/enwiki/wiki/Right_ascension" title="Right ascension">right ascension</a>, governed by combined effects of Earth's axial tilt and <a href="/enwiki/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a>.
</p><p>One can <a href="/enwiki/wiki/Photography" title="Photography">photograph</a> an analemma by keeping a camera at a fixed location and orientation and taking multiple exposures throughout the year, always at the same <a href="/enwiki/wiki/Time_of_day" class="mw-redirect" title="Time of day">time of day</a> (disregarding <a href="/enwiki/wiki/Daylight_saving_time" title="Daylight saving time">daylight saving time</a>, if applicable).
</p><p>Diagrams of analemmas frequently carry marks that show the position of the Sun at various closely spaced dates throughout the year. Analemmas with date marks can be used for various practical purposes.
</p><p>Analemmas (as they are known today) have been used in conjunction with <a href="/enwiki/wiki/Sundial" title="Sundial">sundials</a> since the 18th century to convert between apparent and mean solar time. Before this, the term has a more generic meaning that refers to a graphical procedure of representing <a href="/enwiki/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> objects in <a href="/enwiki/wiki/Two-dimensional_space" title="Two-dimensional space">two dimensions</a>, now known as <a href="/enwiki/wiki/Orthographic_projection" title="Orthographic projection">orthographic projection</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup>
</p><p>Although the term <i>analemma</i> usually refers to Earth's solar analemma, it can be applied to other <a href="/enwiki/wiki/Astronomical_object" title="Astronomical object">celestial bodies</a> as well.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Description"><span class="tocnumber">1</span> <span class="toctext">Description</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#As_seen_from_Earth"><span class="tocnumber">2</span> <span class="toctext">As seen from Earth</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Photography"><span class="tocnumber">3</span> <span class="toctext">Photography</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Calculated_analemmas"><span class="tocnumber">4</span> <span class="toctext">Calculated analemmas</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Estimating_sunrise_and_sunset_data"><span class="tocnumber">5</span> <span class="toctext">Estimating sunrise and sunset data</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Earliest_and_latest_sunrise_and_sunset"><span class="tocnumber">5.1</span> <span class="toctext">Earliest and latest sunrise and sunset</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Times_of_sunrise_and_sunset"><span class="tocnumber">5.2</span> <span class="toctext">Times of sunrise and sunset</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Azimuths_of_sunrise_and_sunset"><span class="tocnumber">5.3</span> <span class="toctext">Azimuths of sunrise and sunset</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-9"><a href="#Seen_from_other_planets"><span class="tocnumber">6</span> <span class="toctext">Seen from other planets</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#Of_geosynchronous_satellites"><span class="tocnumber">7</span> <span class="toctext">Of geosynchronous satellites</span></a></li>
<li class="toclevel-1 tocsection-11"><a href="#Of_quasi-satellites"><span class="tocnumber">8</span> <span class="toctext">Of quasi-satellites</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#See_also"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#References"><span class="tocnumber">10</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2 tocsection-14"><a href="#Footnotes"><span class="tocnumber">10.1</span> <span class="toctext">Footnotes</span></a></li>
<li class="toclevel-2 tocsection-15"><a href="#Citations"><span class="tocnumber">10.2</span> <span class="toctext">Citations</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="#Further_reading"><span class="tocnumber">10.3</span> <span class="toctext">Further reading</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-17"><a href="#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Description">Description</span></h2>
<p>An analemma can be traced by plotting the position of the Sun as viewed from a fixed position on Earth at the same clock time every day for an entire year, or by plotting a graph of the Sun's <a href="/enwiki/wiki/Declination" title="Declination">declination</a> against the <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a>. The resulting curve resembles a long, slender figure-eight with one lobe much larger than the other. This curve is commonly printed on <a href="/enwiki/wiki/Globe#Terrestrial_and_planetary" title="Globe">terrestrial globes</a>, usually in the eastern Pacific Ocean, the only large tropical region with very little land. It is possible, though challenging, to photograph the analemma, by leaving the camera in a fixed position for an entire year and snapping images on 24-hour intervals (or some multiple thereof); see section below.
</p><p>The long axis of the figure—the line segment joining the northernmost point on the analemma to the southernmost—is <a href="/enwiki/wiki/Bisection" title="Bisection">bisected</a> by the <a href="/enwiki/wiki/Celestial_equator" title="Celestial equator">celestial equator</a>, to which it is approximately <a href="/enwiki/wiki/Perpendicular" title="Perpendicular">perpendicular</a>, and has a "length" of twice the <a href="/enwiki/wiki/Axial_tilt" title="Axial tilt">obliquity of the ecliptic</a>, i.e., about 47°. The component along this axis of the Sun's apparent motion is a result of the familiar seasonal variation of the <a href="/enwiki/wiki/Declination" title="Declination">declination</a> of the Sun through the year. The "width" of the figure is due to the equation of time, and its angular extent is the difference between the greatest positive and negative deviations of <a href="/enwiki/wiki/Local_solar_time" class="mw-redirect" title="Local solar time">local solar time</a> from <a href="/enwiki/wiki/Local_mean_time" title="Local mean time">local mean time</a> when this time-difference is related to angle at the rate of 15° per hour, i.e., 360° in 24 h. This width of the analemma is approximately 7.7°, so the length of the figure is more than six times its width. The difference in size of the lobes of the figure-eight form arises mainly from the fact that the <a href="/enwiki/wiki/Perihelion_and_aphelion" class="mw-redirect" title="Perihelion and aphelion">perihelion and aphelion</a> occur far from <a href="/enwiki/wiki/Equinox" title="Equinox">equinoxes</a>. They also occur a mere couple of weeks after <a href="/enwiki/wiki/Solstice" title="Solstice">solstices</a>, which in turn causes slight tilt of the figure eight and its minor lateral asymmetry.
</p><p>There are three parameters that affect the size and shape of the analemma—<a href="/enwiki/wiki/Axial_tilt" title="Axial tilt">obliquity</a>, <a href="/enwiki/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a>, and the angle between the <a href="/enwiki/wiki/Apse_line" title="Apse line">apse line</a> and the line of <a href="/enwiki/wiki/Solstice" title="Solstice">solstices</a>. Viewed from an object with a perfectly circular <a href="/enwiki/wiki/Orbit" title="Orbit">orbit</a> and no axial tilt, the Sun would always appear at the same point in the sky at the same time of day throughout the year and the analemma would be a dot. For an object with a circular orbit but significant axial tilt, the analemma would be a figure of eight with northern and southern lobes equal in size. For an object with an eccentric orbit but no axial tilt, the analemma would be a straight east–west line along the celestial equator.
</p><p>The north–south component of the analemma shows the <a href="/enwiki/wiki/Position_of_the_Sun#Declination_of_the_Sun_as_seen_from_Earth" title="Position of the Sun">Sun's declination</a>, its latitude on the celestial sphere, or the latitude on the Earth at which the Sun is directly overhead. The east–west component shows the <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a>, or the difference between <a href="/enwiki/wiki/Solar_time" title="Solar time">solar time</a> and <a href="/enwiki/wiki/Local_mean_time" title="Local mean time">local mean time</a>. This can be interpreted as how "fast" or "slow" the Sun (or an <a href="/enwiki/wiki/Analemmatic_sundial" title="Analemmatic sundial">analemmatic sundial</a>) is compared to clock time. It also shows how far west or east the Sun is, compared with its mean position. The analemma can be considered as a graph in which the Sun's declination and the equation of time are plotted against each other. In many diagrams of the analemma, a third dimension, that of time, is also included, shown by marks that represent the position of the Sun at various, fairly closely spaced, dates throughout the year.
</p><p>In diagrams, the analemma is drawn as it would be seen in the sky by an observer looking upward. If north is at the top, <i>west</i> is to the <i>right</i>. This corresponds with the sign of the equation of time, which is positive in the westward direction. The further west the Sun is, compared with its mean position, the more "fast" a sundial is, compared with a clock. (See <a href="/enwiki/wiki/Equation_of_time#Sign_of_the_equation_of_time" title="Equation of time">Equation of time#Sign of the equation of time</a>.) If the analemma is a graph with positive declination (north) plotted upward, positive equation of time (west) is plotted to the right. This is the conventional orientation for graphs. When the analemma is marked on a geographical globe, west in the analemma is to the right, while the geographical features on the globe are shown with west to the left. To avoid this confusion, it has been suggested that analemmas on globes should be printed with west to the left, but this is not done, at least, not frequently. In practice, the analemma is so nearly symmetrical that the shapes of the mirror images are not easily distinguished, but if date markings are present, they go in opposite directions. The Sun moves eastward on the analemma near the solstices. This can be used to tell which way the analemma is printed. See the image above, <a class="external text" href="https:/upwiki/wikipedia/commons/thumb/6/6d/Globenmuseum_Vienna_20091010_479.JPG/1280px-Globenmuseum_Vienna_20091010_479.JPG">at high magnification</a>.
</p><p>An analemma that includes an image of a solar eclipse is called a <b>tutulemma</b>, a term coined by photographers Cenk E. Tezel and <a href="/enwiki/wiki/Tun%C3%A7_Tezel" title="Tunç Tezel">Tunç Tezel</a> based on the Turkish word for eclipse.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup>
</p>
<h2><span class="mw-headline" id="As_seen_from_Earth">As seen from Earth</span></h2>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Analemma_on_earth_globe.png" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/e/ef/Analemma_on_earth_globe.png/220px-Analemma_on_earth_globe.png" decoding="async" width="220" height="220" class="thumbimage" data-file-width="2460" data-file-height="2460" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Analemma_on_earth_globe.png" class="internal" title="Enlarge"></a></div>Analemma on Earth as the position of the sun is directly overhead every 24 hours over one year.</div></div></div>
<div class="thumb tleft"><div class="thumbinner" style="width:332px;"><a href="/enwiki/wiki/File:Analemma_Earth.png" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/a/a6/Analemma_Earth.png/330px-Analemma_Earth.png" decoding="async" width="330" height="247" class="thumbimage" data-file-width="1370" data-file-height="1024" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Analemma_Earth.png" class="internal" title="Enlarge"></a></div>Analemma plotted as seen at noon GMT from the <a href="/enwiki/wiki/Royal_Observatory,_Greenwich" title="Royal Observatory, Greenwich">Royal Observatory, Greenwich</a> (<a href="/enwiki/wiki/Latitude" title="Latitude">latitude</a> 51.48° north, <a href="/enwiki/wiki/Longitude" title="Longitude">longitude</a> 0.0015° west).</div></div></div>
<p>Owing to the tilt of Earth's axis (23.439°) and the Earth's orbital eccentricity, the relative location of the Sun above the horizon is not constant from day to day when observed at the same clock time each day. If the time of observation is not 12:00 noon local mean time, then depending on one's geographical latitude, this loop will be inclined at different angles.
</p><p>The figure on the left is an example of an analemma as seen from the Earth's <a href="/enwiki/wiki/Northern_hemisphere" class="mw-redirect" title="Northern hemisphere">northern hemisphere</a>. It is a plot of the position of the Sun at 12:00 noon at <a href="/enwiki/wiki/Royal_Observatory,_Greenwich" title="Royal Observatory, Greenwich">Royal Observatory, Greenwich</a>, England (<a href="/enwiki/wiki/Latitude" title="Latitude">latitude</a> 51.48°N, <a href="/enwiki/wiki/Longitude" title="Longitude">longitude</a> 0.0015°W) during the year 2006. The horizontal axis is the <a href="/enwiki/wiki/Azimuth" title="Azimuth">azimuth</a> angle in degrees (180° is facing south). The vertical axis is the <a href="/enwiki/wiki/Altitude_(astronomy)" class="mw-redirect" title="Altitude (astronomy)">altitude</a> in degrees above the horizon. The first day of each month is shown in black, and the <a href="/enwiki/wiki/Solstice" title="Solstice">solstices</a> and <a href="/enwiki/wiki/Equinox" title="Equinox">equinoxes</a> are shown in green. It can be seen that the equinoxes occur approximately at altitude <span class="nowrap"><i>φ</i> = 90° − 51.5° = 38.5°</span>, and the solstices occur approximately at altitudes <span class="nowrap"><i>φ</i> ± <i>ε</i></span> where <i>ε </i>is the <a href="/enwiki/wiki/Axial_tilt" title="Axial tilt">axial tilt</a> of the earth, 23.4°. The analemma is plotted with its width highly exaggerated, revealing a slight asymmetry (due to the two-week misalignment between the <a href="/enwiki/wiki/Apsis" title="Apsis">apsides</a> of the Earth's orbit and its <a href="/enwiki/wiki/Solstice" title="Solstice">solstices</a>).
</p><p>The analemma is oriented with the smaller loop appearing north of the larger loop. At the <a href="/enwiki/wiki/North_Pole" title="North Pole">North Pole</a>, the analemma would be completely upright (an 8 with the small loop at the top), and only the top half of it would be visible. Heading south, once south of the <a href="/enwiki/wiki/Arctic_Circle" title="Arctic Circle">Arctic Circle</a>, the entire analemma would become visible. If you see it at noon, it continues to be upright, and rises higher from the horizon as you move south. When you get to the equator, it is directly overhead. As you go further south, it moves toward the northern horizon, and is then seen with the larger loop at the top. If, on the other hand, you looked at the analemma in the early morning or evening, it would start to tilt to one side as you moved southward from the North Pole. By the time you got to the <a href="/enwiki/wiki/Equator" title="Equator">equator</a>, the analemma would be completely horizontal. Then, as you continued to go south, it would continue rotating so that the small loop was beneath the large loop in the sky. Once you crossed the <a href="/enwiki/wiki/Antarctic_Circle" title="Antarctic Circle">Antarctic Circle</a>, the analemma, now nearly completely inverted, would start to disappear, until only 50%, part of the larger loop, was visible from the <a href="/enwiki/wiki/South_Pole" title="South Pole">South Pole</a>.<sup id="cite_ref-scienceblogs_4-0" class="reference"><a href="#cite_note-scienceblogs-4">[4]</a></sup>
</p><p>See <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a> for a more detailed description of the east–west characteristics of the analemma.
</p>
<h2><span class="mw-headline" id="Photography">Photography</span></h2>
<p>The first successful analemma photograph ever made was created in 1978–79 by photographer <a href="/enwiki/wiki/Dennis_di_Cicco" title="Dennis di Cicco">Dennis di Cicco</a> over <a href="/enwiki/wiki/Watertown,_Massachusetts" title="Watertown, Massachusetts">Watertown, Massachusetts</a>. Without moving his camera, he made 44 exposures on a single frame of film, all taken at the same time of day at least a week apart. A foreground image and three <a href="/enwiki/wiki/Long-exposure_photography" title="Long-exposure photography">long-exposure images</a> were also included in the same frame, bringing the total number of exposures to 48.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup>
</p>
<h2><span class="mw-headline" id="Calculated_analemmas">Calculated analemmas</span></h2>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Wreath_of_Analemmas.png" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/9/9a/Wreath_of_Analemmas.png/220px-Wreath_of_Analemmas.png" decoding="async" width="220" height="196" class="thumbimage" data-file-width="1463" data-file-height="1301" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Wreath_of_Analemmas.png" class="internal" title="Enlarge"></a></div>"Wreath of Analemmas". Analemmas calculated at 1-hour apart from each other for the geographic center of the contiguous United States. The gray part indicates it is nighttime.</div></div></div>
<p>While photographing analemmas may face technical and practical challenges, they could be calculated conveniently and presented in 3D plots for any given location on the surface of the Earth.<sup id="cite_ref-Zhangetal_6-0" class="reference"><a href="#cite_note-Zhangetal-6">[6]</a></sup>
</p><p>The idea is based on the unit vector with its origin fixed at a chosen point on the surface of the Earth and its direction pointing to the center of the Sun all the time. If we calculate the position of the Sun, namely, the <a href="/enwiki/wiki/Solar_zenith_angle" title="Solar zenith angle">solar zenith angle</a> and <a href="/enwiki/wiki/Solar_azimuth_angle" title="Solar azimuth angle">solar azimuth angle</a> at say, one-hour step, for an entire year, the head of the unit vector traces out 24 analemmas on the unit sphere centered on the chosen point, and this unit sphere is equivalent to the <a href="/enwiki/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a>. The figure on the right is the "wreath of analemmas" calculated for the geographic center of the contiguous United States.
</p>
<div class="thumb tleft"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Analemma_EoT_vs_Delta_2020.png" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/7/70/Analemma_EoT_vs_Delta_2020.png/220px-Analemma_EoT_vs_Delta_2020.png" decoding="async" width="220" height="160" class="thumbimage" data-file-width="1242" data-file-height="904" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Analemma_EoT_vs_Delta_2020.png" class="internal" title="Enlarge"></a></div>Analemma: Equation of time vs. declination of the Sun. Calculated for the year 2020 using the formulas from <i>The <a href="/enwiki/wiki/Astronomical_Almanac" title="Astronomical Almanac">Astronomical Almanac</a> for the Year 2019</i>.</div></div></div>
<p>As often seen on a globe, the analemma is also often plotted as a two-dimensional figure of <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a> vs. declination of the Sun. The adjacent figure ("Analemma: Equation of time...") is calculated using the algorithm presented in the reference<sup id="cite_ref-Zhangetal_6-1" class="reference"><a href="#cite_note-Zhangetal-6">[6]</a></sup> that uses the formulas given in <i>The <a href="/enwiki/wiki/Astronomical_Almanac" title="Astronomical Almanac">Astronomical Almanac</a> for the Year 2019</i>.
</p>
<h2><span class="mw-headline" id="Estimating_sunrise_and_sunset_data">Estimating sunrise and sunset data</span></h2>
<p>If marked to show the position of the Sun on it at fairly regular intervals (such as the 1st, 11th, and 21st days of every <a href="/enwiki/wiki/Calendar_month" class="mw-redirect" title="Calendar month">calendar month</a>) the analemma summarises the apparent motion of the Sun, relative to its mean position, throughout the <a href="/enwiki/wiki/Tropical_year" title="Tropical year">year</a>. A date-marked diagram of the analemma, with equal scales in both <a href="/enwiki/wiki/North" title="North">north</a>–<a href="/enwiki/wiki/South" title="South">south</a> and <a href="/enwiki/wiki/East" title="East">east</a>–<a href="/enwiki/wiki/West" title="West">west</a> directions, can be used as a tool to estimate quantities such as the times of <a href="/enwiki/wiki/Sunrise" title="Sunrise">sunrise</a> and <a href="/enwiki/wiki/Sunset" title="Sunset">sunset</a>, which depend on the Sun's position. Generally, making these estimates depends on visualizing the analemma as a rigid structure in the sky, which moves around the Earth at constant speed so it rises and sets once a day, with the Sun slowly moving around it once a year.
</p><p>Some approximations are involved in the process, chiefly the use of a plane diagram to represent things on the celestial sphere, and the use of drawing and measurement instead of numerical calculation. Because of these, the estimates are not perfectly precise, but they are usually good enough for practical purposes. Also, they have instructional value, showing in a simple visual way how the times of sunrises and sunsets vary.
</p>
<h3><span class="mw-headline" id="Earliest_and_latest_sunrise_and_sunset">Earliest and latest sunrise and sunset</span></h3>
<div class="thumb tright"><div class="thumbinner" style="width:312px;"><a href="/enwiki/wiki/File:Analemma_pattern_in_the_sky.jpg" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/7/74/Analemma_pattern_in_the_sky.jpg/310px-Analemma_pattern_in_the_sky.jpg" decoding="async" width="310" height="251" class="thumbimage" data-file-width="894" data-file-height="724" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Analemma_pattern_in_the_sky.jpg" class="internal" title="Enlarge"></a></div>Diagram of an analemma looking east in the <a href="/enwiki/wiki/Northern_Hemisphere" title="Northern Hemisphere">Northern Hemisphere</a>. The dates of the Sun's position are shown. This analemma is calculated for 9am, not photographed.</div></div></div>
<p>The analemma can be used to find the dates of the earliest and latest <a href="/enwiki/wiki/Sunrise" title="Sunrise">sunrises</a> and <a href="/enwiki/wiki/Sunset" title="Sunset">sunsets</a> of the year. These do not occur on the dates of the <a href="/enwiki/wiki/Solstice" title="Solstice">solstices</a>.
</p><p>With reference to the image of a simulated analemma in the eastern sky, the lowest point of the analemma has just risen above the horizon. If the Sun were at that point, sunrise would have just occurred. This would be the latest sunrise of the year, since all other points on the analemma would rise earlier. Therefore, the date of the latest sunrise is when the Sun is at this lowest point (29 December, when the analemma is tilted as seen from latitude 50° north, as is shown in the diagram); however, in some areas that use <a href="/enwiki/wiki/Daylight_saving_time" title="Daylight saving time">daylight saving time</a>, the date of the latest sunrise occurs on the day before daylight saving time ends. Similarly, when the Sun is at the highest point on the analemma, near its top-left end, (on 15 June) the earliest sunrise of the year will occur. Likewise, at sunset, the earliest sunset will occur when the Sun is at its lowest point on the analemma when it is close to the western horizon, and the latest sunset when it is at the highest point.
</p><p>None of these points is exactly at one of the ends of the analemma, where the Sun is at a solstice. As seen from northern <a href="/enwiki/wiki/Middle_latitude" class="mw-redirect" title="Middle latitude">middle latitudes</a>, as the diagram shows, the earliest sunset occurs some time before the December solstice – typically a week or two before it – and the latest sunrise happens a week or two after the solstice. Thus, the darkest evening occurs in early to mid-December, but the mornings keep getting darker until about the New Year.
</p>
<div class="thumb tleft"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Sunrise_-_Libreville,_Gabon_-_2008.svg" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/a/a7/Sunrise_-_Libreville%2C_Gabon_-_2008.svg/220px-Sunrise_-_Libreville%2C_Gabon_-_2008.svg.png" decoding="async" width="220" height="165" class="thumbimage" srcset="/upwiki/wikipedia/commons/thumb/a/a7/Sunrise_-_Libreville%2C_Gabon_-_2008.svg/330px-Sunrise_-_Libreville%2C_Gabon_-_2008.svg.png 1.5x, /upwiki/wikipedia/commons/thumb/a/a7/Sunrise_-_Libreville%2C_Gabon_-_2008.svg/440px-Sunrise_-_Libreville%2C_Gabon_-_2008.svg.png 2x" data-file-width="1000" data-file-height="750" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Sunrise_-_Libreville,_Gabon_-_2008.svg" class="internal" title="Enlarge"></a></div>Graph of time of sunrise for <a href="/enwiki/wiki/Libreville" title="Libreville">Libreville</a>, <a href="/enwiki/wiki/Gabon" title="Gabon">Gabon</a>, which is very near the <a href="/enwiki/wiki/Equator" title="Equator">Equator</a>. Note there are two maxima and two minima.</div></div></div>
<p>The exact dates are those on which the Sun is at the points where the horizon is <a href="/enwiki/wiki/Tangent" title="Tangent">tangential</a> to the analemma, which in turn depend on how much the analemma, or the north–south meridian passing through it, is tilted from the vertical. This angle of tilt is essentially the co-latitude (90° minus the latitude) of the observer. Calculating these dates numerically is complex, but they can be estimated fairly accurately by placing a straight-edge, tilted at the appropriate angle, tangential to a diagram of the analemma, and reading the dates (interpolating as necessary) when the Sun is at the positions of contact.
</p><p>In <a href="/enwiki/wiki/Middle_latitude" class="mw-redirect" title="Middle latitude">middle latitudes</a>, the dates get further from the solstices as the absolute value of the latitude decreases. In near-equatorial latitudes, the situation is more complex. The analemma lies almost horizontal, so the horizon can be tangential to it at two points, one in each loop of the analemma. Thus there are two widely separated dates in the year when the Sun rises earlier than on adjoining dates, and so on.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">[7]</a></sup>
</p>
<h3><span class="mw-headline" id="Times_of_sunrise_and_sunset">Times of sunrise and sunset</span></h3>
<p>A similar geometrical method, based on the analemma, can be used to find the times of <a href="/enwiki/wiki/Sunrise" title="Sunrise">sunrise</a> and <a href="/enwiki/wiki/Sunset" title="Sunset">sunset</a> at any place on Earth (except within or near the <a href="/enwiki/wiki/Arctic_Circle" title="Arctic Circle">Arctic Circle</a> or <a href="/enwiki/wiki/Antarctic_Circle" title="Antarctic Circle">Antarctic Circle</a>), on any date.
</p><p>The <a href="/enwiki/wiki/Origin_of_coordinates" class="mw-redirect" title="Origin of coordinates">origin</a> of the analemma, where the solar <a href="/enwiki/wiki/Declination" title="Declination">declination</a> and the <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a> are both zero, rises and sets at 6 a.m. and 6 p.m. <a href="/enwiki/wiki/Local_mean_time" title="Local mean time">local mean time</a> on every day of the year, irrespective of the observer's <a href="/enwiki/wiki/Latitude" title="Latitude">latitude</a>. (This estimation does not take account of <a href="/enwiki/wiki/Atmospheric_refraction" title="Atmospheric refraction">atmospheric refraction</a>.) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude (as described above), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date (interpolating between the date markings as necessary), then at sunrise this line represents the horizon.
</p><p>The origin <a href="/enwiki/wiki/Diurnal_motion" title="Diurnal motion">appears to move</a> along the <a href="/enwiki/wiki/Celestial_equator" title="Celestial equator">celestial equator</a> at a speed of 15° per hour, the speed of the <a href="/enwiki/wiki/Earth%27s_rotation" title="Earth's rotation">Earth's rotation</a>. The distance along the celestial equator from the point where it intersects the horizon to the position of the origin of the analemma at sunrise is the distance the origin moves between 6 a.m. and the time of sunrise on the given date. Measuring the length of this equatorial segment therefore gives the difference between 6 a.m. and the time of sunrise.
</p><p>The measurement should, of course, be done on the diagram, but it should be expressed in terms of the angle that would be subtended at an observer on the ground by the corresponding distance in the analemma in the sky. It can be useful to compare it with the length of the analemma, which subtends 47°. Thus, for example, if the length of the equatorial segment on the diagram is 0.4 times the length of the analemma on the diagram, then the segment in the celestial analemma would subtend 0.4 × 47° = 18.8° at the observer on the ground. The angle, in degrees, should be divided by 15 to get the time difference in hours between sunrise and 6 a.m. The sign of the difference is clear from the diagram. If the horizon line at sunrise passes above the origin of the analemma, the Sun rises before 6 a.m., and <i>vice versa</i>.
</p><p>The same technique can be used, <i><a href="/enwiki/wiki/Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a></i>, to estimate the time of sunset. Note that the estimated times are in local mean time. Corrections must be applied to convert them to <a href="/enwiki/wiki/Standard_time" title="Standard time">standard time</a> or <a href="/enwiki/wiki/Daylight_saving_time" title="Daylight saving time">daylight saving time</a>. These corrections will include a term that involves the observer's <a href="/enwiki/wiki/Longitude" title="Longitude">longitude</a>, so both the latitude and longitude affect the final result.
</p>
<h3><span class="mw-headline" id="Azimuths_of_sunrise_and_sunset">Azimuths of sunrise and sunset</span></h3>
<p>The <a href="/enwiki/wiki/Azimuth" title="Azimuth">azimuths</a> (true <a href="/enwiki/wiki/Compass" title="Compass">compass</a> bearings) of the points on the <a href="/enwiki/wiki/Horizon" title="Horizon">horizon</a> where the Sun rises and sets can be easily estimated, using the same diagram as is used to find the times of <a href="/enwiki/wiki/Sunrise" title="Sunrise">sunrise</a> and <a href="/enwiki/wiki/Sunset" title="Sunset">sunset</a>, as described above.
</p><p>The point where the horizon intersects the <a href="/enwiki/wiki/Celestial_equator" title="Celestial equator">celestial equator</a> represents due east or west. The point where the Sun is at sunrise or sunset represents the direction of sunrise or sunset. Simply measuring the distance along the horizon between these points, in angular terms (comparing it with the length of the analemma, as described above), gives the angle between due east or west and the direction of sunrise or sunset. Whether the sunrise or sunset is north or south of due east or west is clear from the diagram. The larger loop of the analemma is at its southern end.
</p>
<h2><span class="mw-headline" id="Seen_from_other_planets">Seen from other planets</span></h2>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/enwiki/wiki/File:Mars_analemma.GIF" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/b/be/Mars_analemma.GIF/220px-Mars_analemma.GIF" decoding="async" width="220" height="306" class="thumbimage" data-file-width="288" data-file-height="400" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Mars_analemma.GIF" class="internal" title="Enlarge"></a></div>An analemma as viewed from <a href="/enwiki/wiki/Mars" title="Mars">Mars</a></div></div></div>
<p>On Earth, the analemma appears as a <a href="https://en.wiktionary.org/wiki/figure_eight" class="extiw" title="wikt:figure eight">figure-eight</a>, but on other <a href="/enwiki/wiki/Solar_System" title="Solar System">Solar System</a> bodies, it may be very different<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">[8]</a></sup> due to the interplay between the three parameters determining the analemma: <a href="/enwiki/wiki/Axial_tilt" title="Axial tilt">axial tilt</a> of each body, <a href="/enwiki/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a> of the body's <a href="/enwiki/wiki/Elliptic_orbit" title="Elliptic orbit">elliptic orbit</a>, and position of either apses or equinoxes. Thus, if either of these variables (such as eccentricity) always dominates the other (as is the case on <a href="/enwiki/wiki/Mars" title="Mars">Mars</a>), the analemma will resemble a <a href="/enwiki/wiki/Drop_(liquid)" title="Drop (liquid)">teardrop</a>. If either of the variables (such as eccentricity) is significant, and the other is practically zero (as is the case on <a href="/enwiki/wiki/Jupiter" title="Jupiter">Jupiter</a>, with only a 3° tilt), the figure will be something much closer to an <a href="/enwiki/wiki/Ellipse" title="Ellipse">ellipse</a>. If both are important enough, that sometimes eccentricity or axial tilt dominates, a figure-eight results.<sup id="cite_ref-scienceblogs_4-1" class="reference"><a href="#cite_note-scienceblogs-4">[4]</a></sup><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/enwiki/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="reference does not exist, the statements are not supported, and contradicted by calculations (January 2018)">citation needed</span></a></i>]</sup>
</p>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><div id="mwe_player_0" class="PopUpMediaTransform" style="width:220px;" videopayload="<div class="mediaContainer" style="width:564px"><video id="mwe_player_1" poster="/upwiki/wikipedia/commons/thumb/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/564px--Mars_Analemma_Time_Lapse_Opportunity.webm.jpg" controls="" preload="none" autoplay="" style="width:564px;height:480px" class="kskin" data-durationhint="48.033" data-startoffset="0" data-mwtitle="Mars_Analemma_Time_Lapse_Opportunity.webm" data-mwprovider="wikimediacommons"><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.480p.vp9.webm" type="video/webm; codecs=&quot;vp9, opus&quot;" data-title="SD VP9 (480P)" data-shorttitle="VP9 480P" data-transcodekey="480p.vp9.webm" data-width="564" data-height="480" data-bandwidth="861528" data-framerate="30"/><source src="/upwiki/wikipedia/commons/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm" type="video/webm; codecs=&quot;vp8&quot;" data-title="Original WebM file, 640 × 544 (984 kbps)" data-shorttitle="WebM source" data-width="640" data-height="544" data-bandwidth="983650" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.480p.webm" type="video/webm; codecs=&quot;vp8, vorbis&quot;" data-title="SD WebM (480P)" data-shorttitle="WebM 480P" data-transcodekey="480p.webm" data-width="564" data-height="480" data-bandwidth="1009504" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.120p.vp9.webm" type="video/webm; codecs=&quot;vp9, opus&quot;" data-title="Lowest bandwidth VP9 (120P)" data-shorttitle="VP9 120P" data-transcodekey="120p.vp9.webm" data-width="142" data-height="120" data-bandwidth="121640" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.160p.webm" type="video/webm; codecs=&quot;vp8, vorbis&quot;" data-title="Low bandwidth WebM (160P)" data-shorttitle="WebM 160P" data-transcodekey="160p.webm" data-width="188" data-height="160" data-bandwidth="127720" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.180p.vp9.webm" type="video/webm; codecs=&quot;vp9, opus&quot;" data-title="Low bandwidth VP9 (180P)" data-shorttitle="VP9 180P" data-transcodekey="180p.vp9.webm" data-width="212" data-height="180" data-bandwidth="201168" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.240p.webm" type="video/webm; codecs=&quot;vp8, vorbis&quot;" data-title="Small WebM (240P)" data-shorttitle="WebM 240P" data-transcodekey="240p.webm" data-width="282" data-height="240" data-bandwidth="255432" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.240p.vp9.webm" type="video/webm; codecs=&quot;vp9, opus&quot;" data-title="Small VP9 (240P)" data-shorttitle="VP9 240P" data-transcodekey="240p.vp9.webm" data-width="282" data-height="240" data-bandwidth="305000" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.360p.webm" type="video/webm; codecs=&quot;vp8, vorbis&quot;" data-title="WebM (360P)" data-shorttitle="WebM 360P" data-transcodekey="360p.webm" data-width="424" data-height="360" data-bandwidth="505976" data-framerate="30"/><source src="/upwiki/wikipedia/commons/transcoded/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/Mars_Analemma_Time_Lapse_Opportunity.webm.360p.vp9.webm" type="video/webm; codecs=&quot;vp9, opus&quot;" data-title="VP9 (360P)" data-shorttitle="VP9 360P" data-transcodekey="360p.vp9.webm" data-width="424" data-height="360" data-bandwidth="506024" data-framerate="30"/></video></div>"><img alt="File:Mars Analemma Time Lapse Opportunity.webm" style="width:220px;height:187px" src="/upwiki/wikipedia/commons/thumb/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm/220px--Mars_Analemma_Time_Lapse_Opportunity.webm.jpg" /><a href="/enwiki//upload.wikimedia.org/wikipedia/commons/1/14/Mars_Analemma_Time_Lapse_Opportunity.webm" title="Play media" target="new"><span class="play-btn-large"><span class="mw-tmh-playtext">Play media</span></span></a></div> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Mars_Analemma_Time_Lapse_Opportunity.webm" class="internal" title="Enlarge"></a></div>A <a href="/enwiki/wiki/Time-lapse" class="mw-redirect" title="Time-lapse">time-lapse</a> of an <a class="mw-selflink selflink">analemma</a> on <a href="/enwiki/wiki/Mars" title="Mars">Mars</a>. Created using images of the <a href="/enwiki/wiki/MarsDial" title="MarsDial">MarsDial</a> on the <i><a href="/enwiki/wiki/Opportunity_(rover)" title="Opportunity (rover)">Opportunity</a></i> rover.</div></div></div>
<p>In the following list, <i>day</i> and <i>year</i> refer to the <a href="/enwiki/wiki/Synodic_day" title="Synodic day">synodic day</a> and <a href="/enwiki/wiki/Sidereal_year" title="Sidereal year">sidereal year</a> of the particular body:
</p>
<dl><dt><a href="/enwiki/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a></dt>
<dd>Because <a href="/enwiki/wiki/Orbital_resonance" title="Orbital resonance">orbital resonance</a> makes the day exactly two years long, the method of plotting the Sun's position at the same time each day would yield only a single point. However, the <a href="/enwiki/wiki/Equation_of_time" title="Equation of time">equation of time</a> can still be calculated for any time of the year, so an analemma can be graphed with this information. The resulting curve is a nearly straight east–west line.</dd>
<dt><a href="/enwiki/wiki/Venus" title="Venus">Venus</a></dt>
<dd>There are slightly less than two days per year, so it would take several years to accumulate a complete analemma by the usual method. The resulting curve is an ellipse.</dd>
<dt><a href="/enwiki/wiki/Mars" title="Mars">Mars</a></dt>
<dd>Teardrop.</dd>
<dt><a href="/enwiki/wiki/Jupiter" title="Jupiter">Jupiter</a></dt>
<dd>Ellipse.</dd>
<dt><a href="/enwiki/wiki/Saturn" title="Saturn">Saturn</a></dt>
<dd>Technically a figure-eight, but the northern loop is so small that it more closely resembles a teardrop.</dd>
<dt><a href="/enwiki/wiki/Uranus" title="Uranus">Uranus</a></dt>
<dd>Figure-eight. (Uranus is tilted past sideways to an angle of 98°. Its orbit is about as eccentric as Jupiter's and more eccentric than Earth's.)</dd>
<dt><a href="/enwiki/wiki/Neptune" title="Neptune">Neptune</a></dt>
<dd>Figure-eight.</dd></dl>
<div style="clear:both;"></div>
<h2><span class="mw-headline" id="Of_geosynchronous_satellites">Of geosynchronous satellites</span></h2>
<div class="thumb tright"><div class="thumbinner" style="width:172px;"><a href="/enwiki/wiki/File:Qzss-45-0.09.jpg" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/c/c2/Qzss-45-0.09.jpg/170px-Qzss-45-0.09.jpg" decoding="async" width="170" height="285" class="thumbimage" data-file-width="271" data-file-height="454" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Qzss-45-0.09.jpg" class="internal" title="Enlarge"></a></div>Groundtrack of <a href="/enwiki/wiki/QZSS" class="mw-redirect" title="QZSS">QZSS</a> geosynchronous orbit. Seen from the ground, its analemma would have a similar shape.</div></div></div>
<p><a href="/enwiki/wiki/Geosynchronous_satellite" title="Geosynchronous satellite">Geosynchronous satellites</a> revolve around the Earth with a period of one <a href="/enwiki/wiki/Sidereal_day" class="mw-redirect" title="Sidereal day">sidereal day</a>. Seen from a fixed point on the Earth's surface, they trace paths in the sky which repeat every day, and are therefore simple and meaningful analemmas. They are generally roughly elliptical, teardrop shaped, or figure-8 in shape. Their shapes and dimensions depend on the parameters of the orbits. A subset of geosynchronous satellites are <a href="/enwiki/wiki/Geostationary_satellites" class="mw-redirect" title="Geostationary satellites">geostationary ones</a>, which ideally have perfectly circular orbits, exactly in the Earth's equatorial plane. A geostationary satellite therefore ideally remains stationary relative to the Earth's surface, staying over a single point on the equator. No real satellite is exactly geostationary, so real ones trace small analemmas in the sky. Since the sizes of the orbits of geosynchronous satellites are similar to the size of the Earth, substantial <a href="/enwiki/wiki/Parallax" title="Parallax">parallax</a> occurs, depending on the location of the observer on the Earth's surface, so observers in different places see different analemmas.
</p><p>The paraboloidal dishes that are used for radio communication with geosynchronous satellites often have to move so as to follow the satellite's daily movement around its analemma. The mechanisms that drive them must therefore be programmed with the parameters of the analemma. Exceptions are dishes that are used with (approximately) geostationary satellites, since these satellites appear to move so little that a fixed dish can function adequately at all times.
</p>
<div class="thumb tleft"><div class="thumbinner" style="width:172px;"><a href="/enwiki/wiki/File:Quasi-satellite_diagram.png" class="image"><img alt="" src="/upwiki/wikipedia/commons/thumb/d/d4/Quasi-satellite_diagram.png/170px-Quasi-satellite_diagram.png" decoding="async" width="170" height="247" class="thumbimage" data-file-width="353" data-file-height="512" /></a> <div class="thumbcaption"><div class="magnify"><a href="/enwiki/wiki/File:Quasi-satellite_diagram.png" class="internal" title="Enlarge"></a></div>Orbital diagram of a quasi-satellite</div></div></div>
<h2><span class="mw-headline" id="Of_quasi-satellites">Of quasi-satellites</span></h2>
<p>A <a href="/enwiki/wiki/Quasi-satellite" title="Quasi-satellite">quasi-satellite</a>, such as the one shown in this diagram, moves in a <a href="/enwiki/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">prograde</a> orbit around the Sun, with the same orbital period (which we will call a year) as the planet it accompanies, but with a different (usually greater) orbital eccentricity. It appears, when seen from the planet, to revolve around the planet once a year in the retrograde direction, but at varying speed and probably not in the ecliptic plane. Relative to its mean position, moving at constant speed in the ecliptic, the quasi-satellite traces an analemma in the planet's sky, going around it once a year.<sup id="cite_ref-analemma_9-0" class="reference"><a href="#cite_note-analemma-9">[9]</a></sup>
</p>
<div style="clear:both;"></div>
<h2><span class="mw-headline" id="See_also">See also</span></h2>
<ul><li><i><a href="/enwiki/wiki/Anathem" title="Anathem">Anathem</a></i></li>
<li><a href="/enwiki/wiki/Armillary_sphere" title="Armillary sphere">Armillary sphere</a></li>
<li><i><a href="/enwiki/wiki/De_architectura" title="De architectura">De architectura</a></i></li>
<li><a href="/enwiki/wiki/Epicycle" class="mw-redirect" title="Epicycle">Epicycle</a></li>
<li><a href="/enwiki/wiki/Lemniscate" title="Lemniscate">Lemniscate</a></li>
<li><i><a href="/enwiki/wiki/On_the_Dioptra" class="mw-redirect" title="On the Dioptra">On the Dioptra</a></i></li></ul>
<h2><span class="mw-headline" id="References">References</span></h2>
<h3><span class="mw-headline" id="Footnotes">Footnotes</span></h3>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha">
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<h3><span class="mw-headline" id="Citations">Citations</span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1011085734"/><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r999302996">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("/upwiki/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("/upwiki/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("/upwiki/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("/upwiki/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFSawyer1994" class="citation journal cs1">Sawyer, Frederick (June 1994). <a rel="nofollow" class="external text" href="https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1994/Bull094-2.pdf#page=2">"Of Analemmas, Mean Time and the Analemmatic Sundial - Part 1"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the British Sundial Society</i>. <b>6</b> (2): 2–6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+British+Sundial+Society&rft.atitle=Of+Analemmas%2C+Mean+Time+and+the+Analemmatic+Sundial+-+Part+1&rft.volume=6&rft.issue=2&rft.pages=2-6&rft.date=1994-06&rft.aulast=Sawyer&rft.aufirst=Frederick&rft_id=https%3A%2F%2Fsundialsoc.org.uk%2Fwp-content%2Fuploads%2Fbulletins%2FB1994%2FBull094-2.pdf%23page%3D2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFSawyer1995" class="citation journal cs1">Sawyer, Frederick (February 1995). <a rel="nofollow" class="external text" href="https://sundialsoc.org.uk/wp-content/uploads/bulletins/B1995/Bull095-1.pdf#page=39">"Of Analemmas, Mean Time and the Analemmatic Sundial - Part 2"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the British Sundial Society</i>. <b>7</b> (1): 39–44.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+British+Sundial+Society&rft.atitle=Of+Analemmas%2C+Mean+Time+and+the+Analemmatic+Sundial+-+Part+2&rft.volume=7&rft.issue=1&rft.pages=39-44&rft.date=1995-02&rft.aulast=Sawyer&rft.aufirst=Frederick&rft_id=https%3A%2F%2Fsundialsoc.org.uk%2Fwp-content%2Fuploads%2Fbulletins%2FB1995%2FBull095-1.pdf%23page%3D39&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFNemiroffBonnell2009" class="citation web cs1">Nemiroff, R.; Bonnell, J., eds. (20 December 2009). <a rel="nofollow" class="external text" href="https://apod.nasa.gov/apod/ap091220.html">"Tutulemma: Solar Eclipse Analemma"</a>. <i><a href="/enwiki/wiki/Astronomy_Picture_of_the_Day" title="Astronomy Picture of the Day">Astronomy Picture of the Day</a></i>. <a href="/enwiki/wiki/NASA" title="NASA">NASA</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Astronomy+Picture+of+the+Day&rft.atitle=Tutulemma%3A+Solar+Eclipse+Analemma&rft.date=2009-12-20&rft_id=https%3A%2F%2Fapod.nasa.gov%2Fapod%2Fap091220.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
</li>
<li id="cite_note-scienceblogs-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-scienceblogs_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-scienceblogs_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php">Why Our Analemma Looks like a Figure 8</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120117120418/http://scienceblogs.com/startswithabang/2009/08/why_our_analemma_looks_like_a.php">Archived</a> January 17, 2012, at the <a href="/enwiki/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://petapixel.com/2011/09/20/more-people-have-walked-on-the-moon-than-have-captured-the-analemma/">"More People Have Walked on the Moon Than Have Captured the Analemma"</a>. <i>PetaPixel</i>. 20 September 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-07-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PetaPixel&rft.atitle=More+People+Have+Walked+on+the+Moon+Than+Have+Captured+the+Analemma&rft.date=2011-09-20&rft_id=https%3A%2F%2Fpetapixel.com%2F2011%2F09%2F20%2Fmore-people-have-walked-on-the-moon-than-have-captured-the-analemma%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span> Includes image of original 1979 publication.</span>
</li>
<li id="cite_note-Zhangetal-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Zhangetal_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zhangetal_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFZhangStackhouseMacphersonMikovitz2021" class="citation journal cs1">Zhang, Taiping; Stackhouse, Paul W.; Macpherson, Bradley; Mikovitz, J. Colleen (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.renene.2021.03.047">"A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function"</a>. <i>Renewable Energy</i>. Elsevier BV. <b>172</b>: 1333–1340. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.renene.2021.03.047">10.1016/j.renene.2021.03.047</a></span>. <a href="/enwiki/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="/enwiki//www.worldcat.org/issn/0960-1481">0960-1481</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Renewable+Energy&rft.atitle=A+solar+azimuth+formula+that+renders+circumstantial+treatment+unnecessary+without+compromising+mathematical+rigor%3A+Mathematical+setup%2C+application+and+extension+of+a+formula+based+on+the+subsolar+point+and+atan2+function&rft.volume=172&rft.pages=1333-1340&rft.date=2021&rft_id=info%3Adoi%2F10.1016%2Fj.renene.2021.03.047&rft.issn=0960-1481&rft.aulast=Zhang&rft.aufirst=Taiping&rft.au=Stackhouse%2C+Paul+W.&rft.au=Macpherson%2C+Bradley&rft.au=Mikovitz%2C+J.+Colleen&rft_id=%2F%2Fdoi.org%2F10.1016%252Fj.renene.2021.03.047&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://aa.usno.navy.mil/faq/docs/dark_days.php">"The Dark Days of Winter"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Dark+Days+of+Winter&rft_id=http%3A%2F%2Faa.usno.navy.mil%2Ffaq%2Fdocs%2Fdark_days.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span> at the <a rel="nofollow" class="external text" href="http://www.usno.navy.mil/USNO">USNO website</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160131231447/http://www.usno.navy.mil/USNO">Archived</a> January 31, 2016, at the <a href="/enwiki/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://analemma.com/other-analemmas.html">"Other Analemmas"</a>. <i>analemma.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">24 March</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=analemma.com&rft.atitle=Other+Analemmas&rft_id=https%3A%2F%2Fanalemma.com%2Fother-analemmas.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
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<li id="cite_note-analemma-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-analemma_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFde_la_Fuente_Marcosde_la_Fuente_Marcos2016" class="citation journal cs1">de la Fuente Marcos, Carlos; de la Fuente Marcos, Raúl (2016). "The analemma criterion: accidental quasi-satellites are indeed true quasi-satellites". <i><a href="/enwiki/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society" title="Monthly Notices of the Royal Astronomical Society">Monthly Notices of the Royal Astronomical Society</a></i>. <b>462</b> (3): 3344–3349. <a href="/enwiki/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="/enwiki//arxiv.org/abs/1607.06686">1607.06686</a></span>. <a href="/enwiki/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016MNRAS.462.3344D">2016MNRAS.462.3344D</a>. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstw1833">10.1093/mnras/stw1833</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=The+analemma+criterion%3A+accidental+quasi-satellites+are+indeed+true+quasi-satellites&rft.volume=462&rft.issue=3&rft.pages=3344-3349&rft.date=2016&rft_id=info%3Aarxiv%2F1607.06686&rft_id=info%3Adoi%2F10.1093%2Fmnras%2Fstw1833&rft_id=info%3Abibcode%2F2016MNRAS.462.3344D&rft.aulast=de+la+Fuente+Marcos&rft.aufirst=Carlos&rft.au=de+la+Fuente+Marcos%2C+Ra%C3%BAl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></span>
</li>
</ol></div></div>
<h3><span class="mw-headline" id="Further_reading">Further reading</span></h3>
<style data-mw-deduplicate="TemplateStyles:r1011217839">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}</style><div class="refbegin" style="">
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFOliver1972" class="citation journal cs1">Oliver, Bernard M. (1972). "The Shape of the Analemma". <i>Sky and Telescope</i>. <b>44</b>: 20. <a href="/enwiki/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1972S&T....44...20O">1972S&T....44...20O</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sky+and+Telescope&rft.atitle=The+Shape+of+the+Analemma&rft.volume=44&rft.pages=20&rft.date=1972&rft_id=info%3Abibcode%2F1972S%26T....44...20O&rft.aulast=Oliver&rft.aufirst=Bernard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFKittlerDarula2004" class="citation journal cs1">Kittler, Richard; Darula, Stan (2004). "Analemma, the Ancient Sketch of Fictitious Sunpath Geometry—Sun, Time and History of Mathematics". <i>Architectural Science Review</i>. <b>47</b> (2): 141–4. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00038628.2004.9697037">10.1080/00038628.2004.9697037</a>. <a href="/enwiki/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122005748">122005748</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Architectural+Science+Review&rft.atitle=Analemma%2C+the+Ancient+Sketch+of+Fictitious+Sunpath+Geometry%E2%80%94Sun%2C+Time+and+History+of+Mathematics&rft.volume=47&rft.issue=2&rft.pages=141-4&rft.date=2004&rft_id=info%3Adoi%2F10.1080%2F00038628.2004.9697037&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122005748%23id-name%3DS2CID&rft.aulast=Kittler&rft.aufirst=Richard&rft.au=Darula%2C+Stan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFSidoli2005" class="citation journal cs1">Sidoli, Nathan (2005). "Heron's Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities". <i>Centaurus</i>. <b>47</b> (3): 236–58. <a href="/enwiki/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005Cent...47..236S">2005Cent...47..236S</a>. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1600-0498.2005.470304.x">10.1111/j.1600-0498.2005.470304.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Centaurus&rft.atitle=Heron%27s+Dioptra+35+and+Analemma+Methods%3A+An+Astronomical+Determination+of+the+Distance+between+Two+Cities&rft.volume=47&rft.issue=3&rft.pages=236-58&rft.date=2005&rft_id=info%3Adoi%2F10.1111%2Fj.1600-0498.2005.470304.x&rft_id=info%3Abibcode%2F2005Cent...47..236S&rft.aulast=Sidoli&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFSemazziScroggsPouliotMcKee-Burrows2005" class="citation journal cs1">Semazzi, Fredrick H.M.; Scroggs, Jeffrey S.; Pouliot, George A.; McKee-Burrows, Analemma Leia; Norman, Matthew; Poojary, Vikram; Tsai, Yu-Ming (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.2151%2Fjmsj.83.851">"On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere"</a>. <i>Journal of the Meteorological Society of Japan</i>. <b>83</b> (5): 851–69. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2151%2Fjmsj.83.851">10.2151/jmsj.83.851</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Meteorological+Society+of+Japan&rft.atitle=On+the+Accuracy+of+Semi-Lagrangian+Numerical+Simulation+of+Internal+Gravity+Wave+Motion+in+the+Atmosphere&rft.volume=83&rft.issue=5&rft.pages=851-69&rft.date=2005&rft_id=info%3Adoi%2F10.2151%2Fjmsj.83.851&rft.aulast=Semazzi&rft.aufirst=Fredrick+H.M.&rft.au=Scroggs%2C+Jeffrey+S.&rft.au=Pouliot%2C+George+A.&rft.au=McKee-Burrows%2C+Analemma+Leia&rft.au=Norman%2C+Matthew&rft.au=Poojary%2C+Vikram&rft.au=Tsai%2C+Yu-Ming&rft_id=%2F%2Fdoi.org%2F10.2151%252Fjmsj.83.851&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFLuckey1927" class="citation journal cs1">Luckey, P. (1927). "Das Analemma von Ptolemäus" [The analemma by Ptolemy]. <i>Astronomische Nachrichten</i> (in German). <b>230</b> (2): 17–46. <a href="/enwiki/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1927AN....230...17L">1927AN....230...17L</a>. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fasna.19272300202">10.1002/asna.19272300202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomische+Nachrichten&rft.atitle=Das+Analemma+von+Ptolem%C3%A4us&rft.volume=230&rft.issue=2&rft.pages=17-46&rft.date=1927&rft_id=info%3Adoi%2F10.1002%2Fasna.19272300202&rft_id=info%3Abibcode%2F1927AN....230...17L&rft.aulast=Luckey&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFId1969" class="citation journal cs1">Id, Yusif (December 1969). "An Analemma Construction for Right and Oblique Ascensions". <i>The Mathematics Teacher</i>. <b>62</b> (8): 669–72. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.62.8.0669">10.5951/MT.62.8.0669</a>. <a href="/enwiki/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="/enwiki//www.jstor.org/stable/27958259">27958259</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=An+Analemma+Construction+for+Right+and+Oblique+Ascensions&rft.volume=62&rft.issue=8&rft.pages=669-72&rft.date=1969-12&rft_id=info%3Adoi%2F10.5951%2FMT.62.8.0669&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F27958259%23id-name%3DJSTOR&rft.aulast=Id&rft.aufirst=Yusif&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFYeow2002" class="citation book cs1">Yeow, Teo Shin (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110517111058/http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf"><i>The Analemma for Latitudinally-Challenged People</i></a> <span class="cs1-format">(PDF)</span> (BS Thesis). National University of Singapore. Archived from <a rel="nofollow" class="external text" href="http://www.math.nus.edu.sg/aslaksen/projects/tsy.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-05-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-02-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analemma+for+Latitudinally-Challenged+People&rft.pub=National+University+of+Singapore&rft.date=2002&rft.aulast=Yeow&rft.aufirst=Teo+Shin&rft_id=http%3A%2F%2Fwww.math.nus.edu.sg%2Faslaksen%2Fprojects%2Ftsy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalemma" class="Z3988"></span></li></ul>
</div>
<h2><span class="mw-headline" id="External_links">External links</span></h2>
<table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:#f9f9f9;border:1px solid #aaa;color:#000">
<tbody><tr>
<td class="mbox-image"><img alt="" src="/upwiki/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="noviewer" srcset="/upwiki/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, /upwiki/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></td>
<td class="mbox-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/analemma" class="extiw" title="wiktionary:Special:Search/analemma">analemma</a></b></i> in Wiktionary, the free dictionary.</td></tr>
</tbody></table>
<table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:#f9f9f9;border:1px solid #aaa;color:#000">
<tbody><tr>
<td class="mbox-image"><img alt="" src="/upwiki/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="noviewer" srcset="/upwiki/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, /upwiki/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></td>
<td class="mbox-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Analemma" class="extiw" title="commons:Category:Analemma">Analemma</a></span>.</td></tr>
</tbody></table>
<ul><li><a rel="nofollow" class="external text" href="http://www.perseus.gr/Astro-Solar-Analemma.htm">Analemma Series from Sunrise to Sunset</a></li>
<li><a rel="nofollow" class="external text" href="http://epod.usra.edu/blog/2005/01/colorado-analemma.html">Earth Science Photo of the Day</a> (2005-01-22)</li>
<li><a rel="nofollow" class="external text" href="http://moonkmft.co.uk/EquationOfTime.html">The Equation of Time and the Analemma</a> — by Kieron Taylor</li>
<li><a rel="nofollow" class="external text" href="http://www.nikolasschiller.com/blog/index.php/archives/2008/08/01/1449/">The Use of the Analemma</a> — from an inset from Bowles's New and Accurate Map of the World (1780)</li>
<li><a rel="nofollow" class="external text" href="http://www.astronomycorner.net/games/analemma.html">Figure-Eight in the Sky</a> — contains link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities)</li>
<li><a rel="nofollow" class="external text" href="http://www.analemma.com/">Analemma.com</a> — dedicated to the analemma.</li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060323145857/http://www.wsanford.com/~wsanford/exo/sundials/analemma_calc.html">Calculate and Chart the Analemma</a> — a web site offered by a <a href="/enwiki/wiki/Fairfax_County_Public_Schools" title="Fairfax County Public Schools">Fairfax County Public Schools</a> planetarium that describes the analemma and also offers a downloadable spreadsheet that allows the user to experiment with analemmas of varying shapes.</li>
<li><a rel="nofollow" class="external text" href="http://www.jgiesen.de/analemma/">Analemma Sundial Applet</a> — includes many reference charts.</li>
<li><i><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/Analemmas/">Analemmas</a></i> — by <a href="/enwiki/wiki/Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a> based on a program by Michael Trott, <a href="/enwiki/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li>
<li><i><a rel="nofollow" class="external text" href="http://www.mail-archive.com/sundial@uni-koeln.de/msg11062.html">Analemma in Verse</a></i> by Tad Dunne</li>
<li><i><a rel="nofollow" class="external text" href="http://www.spaceweather.com/glossary/tutulemma.htm">The Making of a Tutulemma</a></i> by <a href="/enwiki/wiki/Tun%C3%A7_Tezel" title="Tunç Tezel">Tunç Tezel</a></li>
<li><i><a rel="nofollow" class="external text" href="http://analemma.pl/english-version">Making of a Solargraphy Analemma</a></i> by <a href="/enwiki/w/index.php?title=Maciej_Zapi%C3%B3r_and_%C5%81ukasz_Fajfrowski&action=edit&redlink=1" class="new" title="Maciej Zapiór and Łukasz Fajfrowski (page does not exist)">Maciej Zapiór and Łukasz Fajfrowski</a></li>
<li><a rel="nofollow" class="external text" href="http://equation-of-time.info">Equation-of-Time.info</a> - a multipage website with many illustrations and videos dedicated to the Equation of Time, its components, its history, how it can be displayed in tables, curves, analemmas, etc., its use to correct sundials, astronomy, clocks, how it can be produced mechanically and much more : by Kevin Karney</li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20191018103653/https://ciechanow.ski/earth-and-sun/">Earth and Sun</a> — an interactive blog post explaining the phenomenon</li>
<li><a href="/enwiki/wiki/Astronomy_Picture_of_the_Day" title="Astronomy Picture of the Day">Astronomy Picture of the Day</a>
<ul><li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap020709.html">2002-07-09</a> — Analemma</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap030320.html">2003-03-20</a> — Sunrise Analemma</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap040621.html">2004-06-21</a> — Analemma over Ancient Nemea</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap050713.html">2005-07-13</a> — Analemma of the Moon</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap061223.html">2006-12-23</a> — Analemma over the Temple of Olympian Zeus</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap061230.html">2006-12-30</a> — Martian Analemma at Sagan Memorial Station (simulated)</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap070617.html">2007-06-17</a> — Analemma over Ukraine</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap071204.html">2007-12-04</a> — Analemma over New Jersey (film)</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap081221.html">2008-12-21</a> — Analemma over the Porch of Maidens</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap091220.html">2009-12-20</a> — Tutulemma: Solar Eclipse Analemma</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap101231.html">2010-12-31</a> — Analemma 2010</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap120920.html">2012-09-20</a> — Sunrise Analemma (with a little extra)</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap131014.html">2013-10-14</a> — High Noon Analemma Over Azerbaijan</li>
<li><a rel="nofollow" class="external text" href="http://antwrp.gsfc.nasa.gov/apod/ap140320.html">2014-03-20</a> — Solargraphy Analemma</li></ul></li></ul>
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<ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://academic.microsoft.com/v2/detail/112969597">Microsoft Academic</a></span></li></ul>
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Whether or not the change was made through a Tor exit node (tor_exit_node) | false |
Unix timestamp of change (timestamp) | 1631723793 |